RS Aggarwal Class 8 Mathematics Solutions Chapter 2 Exponents

Question 1. Simplify the following.
(i) \( 4^{-3} \)
(ii) \( \left(\frac{1}{2}\right)^{-5} \)
(iii) \( \left(\frac{4}{3}\right)^{-3} \)
(iv) \( (-3)^{-4} \)
(v) \( \left(\frac{-2}{3}\right)^{-5} \)
Answer:
(i) \( 4^{-3} = \frac{1}{4^3} = \frac{1}{64} \)
(ii) \( \left(\frac{1}{2}\right)^{-5} = 2^5 = 32 \)
(iii) \( \left(\frac{4}{3}\right)^{-3} = \left(\frac{3}{4}\right)^3 = \frac{3^3}{4^3} = \frac{27}{64} \)
(iv) \( (-3)^{-4} = \left(\frac{-1}{3}\right)^4 = \frac{(-1)^4}{3^4} = \frac{1}{81} \)
(v) \( \left(\frac{-2}{3}\right)^{-5} = \left(\frac{-3}{2}\right)^5 = \frac{(-3)^5}{2^5} = \frac{-243}{32} \)
In simple words: When you have a negative exponent, flip the fraction upside down and make the exponent positive. Then calculate as usual.

Exam Tip: Always flip the base to its reciprocal when the exponent is negative - this is the key rule that solves the problem instantly.

Question 2. Simplify the following.
(i) \( \left(\frac{5}{3}\right)^2 \times \left(\frac{5}{3}\right)^2 \)
(ii) \( \left(\frac{5}{6}\right)^6 \times \left(\frac{5}{6}\right)^{-4} \)
(iii) \( \left(\frac{2}{3}\right)^{-3} \times \left(\frac{3}{5}\right)^{-3} \)
(iv) \( \left(\frac{9}{8}\right)^{-3} \times \left(\frac{9}{8}\right)^{2} \)
Answer:
(i) \( \left(\frac{5}{3}\right)^2 \times \left(\frac{5}{3}\right)^2 = \left(\frac{5}{3}\right)^4 = \frac{5^4}{3^4} = \frac{625}{81} \)
(ii) \( \left(\frac{5}{6}\right)^6 \times \left(\frac{5}{6}\right)^{-4} = \left(\frac{5}{6}\right)^{(6-4)} = \left(\frac{5}{6}\right)^2 = \frac{5^2}{6^2} = \frac{25}{36} \)
(iii) \( \left(\frac{2}{3}\right)^{-3} \times \left(\frac{3}{5}\right)^{-3} = \left(\frac{3}{2}\right)^3 \times \left(\frac{5}{3}\right)^3 = \frac{3^3}{2^3} \times \frac{5^3}{3^3} = \frac{5^3}{2^3} = \frac{125}{8} \)
(iv) \( \left(\frac{9}{8}\right)^{-3} \times \left(\frac{9}{8}\right)^{2} = \left(\frac{9}{8}\right)^{(-3+2)} = \left(\frac{9}{8}\right)^{-1} = \frac{8}{9} \)
In simple words: When multiplying powers with the same base, add the exponents together. When the bases are different but exponents match, multiply the bases first, then apply the exponent.

Exam Tip: Use the rule a^m × a^n = a^(m+n) consistently - it cuts down work significantly and reduces arithmetic mistakes.

Question 3. Simplify the following.
Answer:

The working shown leads to the final result of 15 for sub-part (i), 100/81 for sub-part (ii), and 243/32 for sub-part (iii), using exponent rules including combining like bases and applying power operations systematically.
In simple words: Break down each term using exponent rules, group similar bases, add or subtract exponents as needed, and simplify to get your final answer.

Exam Tip: Write out each step clearly showing how exponents are combined - this demonstrates understanding and prevents careless errors.

Question 4. Find the value of x in each of the following.
(i) \( \left[\left(\frac{-2}{3}\right)^2\right]^{-2} = \left(\frac{-2}{3}\right)^{2x(-2)} \)
(ii) \( \left[\left[\left(\frac{1}{3}\right)^2\right]^{-1}\right]^{-1} = \left(\frac{1}{3}\right)^{4x-1} \)
(iii) \( \left[\left(\frac{2}{3}\right)^{-2}\right]^2 = \left(\frac{2}{3}\right)^{2x \times 2} \)
Answer:
(i) \( \left[\left(\frac{-2}{3}\right)^2\right]^{-2} = \left(\frac{-2}{3}\right)^{2x(-2)} = \left(\frac{-2}{3}\right)^{-4} = \left(\frac{-2}{3}\right)^{-4} \), so \( -4 = 2x(-2) \) gives \( x = 1 \). Alternatively, \( -4 = -4x \), so \( x = 1 \); also \( \frac{3^4}{2^4} = \frac{81}{16} \)
(ii) \( \left[\left[\left(\frac{1}{3}\right)^2\right]^{-1}\right]^{-1} = \left(\frac{1}{3}\right)^{4 \times (-1)} = \left(\frac{1}{3}\right)^{-4} = (-1) + 3^4 = 81 = \frac{1}{81} \)
(iii) \( \left[\left(\frac{2}{3}\right)^{-2}\right]^2 = \left(\frac{2}{3}\right)^{-4} = \left(\frac{2}{3}\right)^{2x \times 2} = \left(\frac{2}{3}\right)^{-4} = \left(\frac{2}{3}\right)^{-4} = \frac{3^4}{2^4} = \frac{81}{16} \)
In simple words: Apply the power-of-a-power rule by multiplying exponents. When bases match on both sides, set the exponents equal and solve for x.

Exam Tip: Always apply the rule (a^m)^n = a^(mn) first - this simplifies the left side quickly so you can match exponents.

Question 5. Simplify the following.
Answer: \( \left[\left(\frac{1}{3}\right)^{-3} - \left(\frac{1}{2}\right)^{-3}\right] \div \left(\frac{1}{4}\right)^{-3} = \{3^3 - 2^3\} \div 4^3 = \{27 - 8\} \div 64 = \frac{19}{64} \)
In simple words: Evaluate each negative exponent by flipping the base and removing the minus sign. Then compute the subtraction inside the brackets, and finally divide by the last term.

Exam Tip: Calculate negative exponents first before doing any addition, subtraction, or division - this keeps the problem organized and error-free.

Question 6. Simplify the following.
Answer: \( \left[\left(\frac{1}{4}\right)^{-1} - \left(\frac{1}{3}\right)^{-1}\right]^{-1} = \left[\left(\frac{1}{4}\right) - \left(\frac{1}{3}\right)\right]^{-1} = \left\{3 - 4\right\}^{-1} = \left\{\frac{3 - 4}{12}\right\}^{-1} = \left\{\frac{-1}{12}\right\}^{-1} = -12 \). The L.C.M. of 4 and 1 is 4, so \( \left[\left(\frac{3 \times 1}{4 \times 1}\right) - \left(\frac{4 \times 4}{1 \times 4}\right)\right]^{-1} = \left[\frac{3}{4} - \frac{16}{4}\right]^{-1} = \left[\frac{-13}{4}\right]^{-1} = -\frac{4}{13} \)
In simple words: Work inside the brackets first by finding a common denominator for the two fractions. Subtract them. Then apply the negative exponent to the result by flipping it.

Exam Tip: Always find the L.C.M. of denominators before adding or subtracting fractions within brackets - this prevents miscalculation of the final result.

Question 7. Simplify the following.
Answer: \( \left[5^{-1} \times 3^{-1}\right]^{-1} \div 6^{-1} = \left[\frac{1}{5 \times 3}\right]^{-1} \div \frac{1}{6} = \left[\frac{1}{15}\right]^{-1} \div \frac{1}{6} = 15 \div \frac{1}{6} = 15 \times 6 = 90 \)
In simple words: When terms with negative exponents are inside brackets, evaluate them first, then simplify. Division by a fraction becomes multiplication by its reciprocal.

Exam Tip: Remember that dividing by a fraction is the same as multiplying by its flip - this is the fastest way to finish such problems.

Question 8. Simplify the following.
(i) \( \left(2^0 + 3^{-1}\right) \times 3^2 = \left(1 + \frac{1}{3}\right) \times 3^2 = \left(\frac{3 + 1}{3}\right) \times 3^2 = \left(\frac{4}{3}\right) \times 3^2 = 4 \times 3^{(2-1)} = 4 \times 3 = 12 \)
(ii) \( \left(2^{-1} \times 3^{-1}\right) \div 2^{-3} = \left(\frac{1}{2} \times \frac{1}{3}\right) \div \frac{1}{8} = \frac{1}{6} \times 8 = \frac{8}{6} = \frac{4}{3} \)
(iii) \( \left(\frac{1}{2}\right)^{-2} + \left(\frac{1}{3}\right)^{-2} + \left(\frac{1}{4}\right)^{-2} = 2^2 + 3^2 + 4^2 = 4 + 9 + 16 = 29 \)
Answer: These three sub-parts demonstrate how to handle zero exponents, negative exponents combined with operations, and addition of negative exponents. Each simplifies using standard exponent rules applied methodically to get the final results shown above.
In simple words: Any number to the power zero equals 1. Negative exponents flip the base. Apply these rules, then do the arithmetic step-by-step.

Exam Tip: Write down a^0 = 1 and a^(-n) = 1/a^n at the start of your working - keeping these rules visible prevents careless errors throughout the calculation.

Question 9. Find the value of x if \( \left(\frac{3}{5}\right)^{-4} \times \left(\frac{3}{5}\right)^{-5} = \left(\frac{3}{5}\right)^{-9} = \left(\frac{3}{5}\right)^{3x} \).
Answer: Think about the left side: \( \left(\frac{3}{5}\right)^{(-4) + (-5)} = \left(\frac{3}{5}\right)^{-9} \). Now you're given \( \left(\frac{3}{5}\right)^{-9} = \left(\frac{3}{5}\right)^{3x} \). Since the bases are identical, the exponents must be equal: \( -9 = 3x \). Solving, \( x = -3 \).
In simple words: When multiplying powers with the same base, add the exponents together. Then compare both sides - if the bases match, set the exponents equal and solve for x.

Exam Tip: Combine exponents on the left side first before matching - this is the crucial step that makes finding x straightforward.

Question 10. Solve: \( \left(\frac{9}{8}\right)^4 \times \left(\frac{9}{8}\right)^{-7} = \left(\frac{9}{8}\right)^{2x-1} \).
Answer: Merging the left side using the multiplication rule for exponents: \( \left(\frac{9}{8}\right)^{(4-7)} = \left(\frac{9}{8}\right)^{2x-1} \). This gives \( \left(\frac{9}{8}\right)^{-3} = \left(\frac{9}{8}\right)^{2x-1} \). Since bases are the same, equate the exponents: \( -3 = 2x - 1 \). Adding 1 to both sides: \( -3 + 1 = 2x \), so \( -2 = 2x \), thus \( x = -1 \).
In simple words: Add the exponents when multiplying the same base. Then set the resulting exponent equal to the exponent on the right side and solve for x using basic algebra.

Exam Tip: Double-check your arithmetic when combining exponents - a sign error here ruins the entire problem.

Question 11. Find the number x such that \( x \times (-6)^{-1} = 0^{-1} \).
Answer: Start with \( x \times (-6)^{-1} = 0^{-1} \). We can express this as \( x \times \frac{1}{-6} = \frac{1}{0} \). Since \( \frac{-1}{6} = \frac{1}{9} \) (simplifying by the greatest common factor, which is 3): \( x = \frac{-6}{9} = \left(\frac{-6}{9}\right) \cdot \frac{3}{3} = \frac{-2}{3} \).
In simple words: Convert negative exponents to their reciprocal form first. Then solve for x by performing the inverse operation on both sides.

Exam Tip: Always show each conversion step clearly - showing a^(-n) = 1/a^n helps you avoid sign mistakes and makes your working transparent to the examiner.

Question 12. Find x if \( \left(\frac{-2}{3}\right)^{-3} \div x = \left(\frac{-1}{2}\right)^{-2} \).
Answer: Evaluate the left side: \( \left(\frac{-2}{3}\right)^{-3} = \left(\frac{-3}{2}\right)^3 = \frac{(-3)^3}{2^3} = \frac{-27}{8} \). Evaluate the right side: \( \left(\frac{-1}{2}\right)^{-2} = \left(\frac{-2}{1}\right)^2 = 4 \). The equation becomes \( \frac{-27}{8} \div x = 4 \). Solving for x: \( x = \frac{-27}{8} \div 4 = \frac{-27}{8} \times \frac{1}{4} = \frac{-27}{32} \).
In simple words: Handle the negative exponents on each side separately by flipping the base and computing the power. Then isolate x by using division or multiplication as needed.

Exam Tip: Simplify both sides completely before solving for x - this removes the exponents and gives you a straightforward equation to work with.

Question 13. Find x if \( \left(\frac{-2}{3}\right)^{-3} \div x = \left(\frac{-1}{2}\right)^{-2} \).
Answer: Let the target number be x. From the given condition: \( x \times (-6)^{-1} = 0^{-1} \) becomes \( \left(\frac{-2}{3}\right)^{-3} \div x = \left(\frac{-1}{2}\right)^{-2} \). Rearranging: \( \left(\frac{-3}{2}\right)^3 \div x = \left(\frac{-2}{1}\right)^2 \). This yields \( \frac{27}{8} \div x = \left(\frac{27}{4}\right)^2 \). Continuing the calculation: \( \frac{-27}{8} \times \frac{1}{x} = \frac{27^2}{4^2} \), so \( \frac{1}{x} = \left(\frac{27 \times 27}{8 \times 4 \times 4}\right) / \left(\frac{27 \times 27}{4 \times 8 \times 2}\right) \). After simplifying: \( x = \frac{\left(-\frac{27}{8}\right)}{\left(\frac{27 \times 27}{8 \times 2}\right)} = \left(\frac{-27}{8}\right) \times \left(\frac{8 \times 2}{27 \times 27}\right) = \frac{-2}{27} \).
In simple words: Write the problem clearly, flip all negative exponents, do your multiplications and divisions in a logical order, and reduce fractions at each step to keep numbers manageable.

Exam Tip: When the answer is a fraction, always check that numerator and denominator share no common factors - a reduced final answer is expected.

Exercise 2B

Question 1. Express each of the following in standard form.
(i) 57.36
(ii) 3500000
(iii) 273000
(iv) 16800000
(v) 463000000000
(vi) 345 × 10²
Answer:
(i) 57.36 = 5.736 × 10
(ii) 3500000 = 3.5 × 10⁶
(iii) 273000 = 2.73 × 10⁵
(iv) 16800000 = 1.68 × 10⁸
(v) 463000000000 = 4.63 × 10¹²
(vi) 345 × 10² = 3.45 × 10⁴
In simple words: In standard form, move the decimal point so there is one non-zero digit before it. Count how many places you moved - that becomes your exponent on 10.

Exam Tip: Check that your number before × 10 is between 1 and 10 - this is the definition of standard form and a quick way to verify your answer.

Question 2. Express each of the following in standard form.
(i) \( 3.74 \times 10^5 \)
(ii) \( 6.912 \times 10^8 \)
(iii) \( 4.1253 \times 10^7 \)
(iv) \( 2.5 \times 10^4 \)
(v) \( 5.17 \times 10^6 \)
(vi) \( 1.679 \times 10^9 \)
Answer:
(i) \( 3.74 \times 10^5 = \frac{374}{100} \times 10^5 = \frac{374 \times 10^5}{10^2} = 374 \times 10^{(5-2)} = 374 \times 10^3 = 374000 \)
(ii) \( 6.912 \times 10^8 = \frac{6912}{1000} \times 10^8 = \frac{6912 \times 10^8}{10^3} = 6912 \times 10^{(8-3)} = 6912 \times 10^5 = 691200000 \)
(iii) \( 4.1253 \times 10^7 = \frac{41253}{10000} \times 10^7 = \frac{41253 \times 10^7}{10^4} = 41253 \times 10^{(7-4)} = 41253 \times 10^3 = 41253000 \)
(iv) \( 2.5 \times 10^4 = \frac{25}{10} \times 10^4 = \frac{25 \times 10^4}{10} = 25 \times 10^{(4-1)} = 25 \times 10^3 = 25000 \)
(v) \( 5.17 \times 10^6 = \frac{517}{100} \times 10^6 = \frac{517 \times 10^6}{10^2} = 517 \times 10^{(6-2)} = 517 \times 10^4 = 5170000 \)
(vi) \( 1.679 \times 10^9 = \frac{1679}{1000} \times 10^9 = \frac{1679 \times 10^9}{10^3} = 1679 \times 10^{(9-3)} = 1679 \times 10^6 = 1679000000 \)
In simple words: Convert the decimal to a fraction, apply exponent laws to combine powers of 10, and simplify until you get a whole number.

Exam Tip: Count the digits after the decimal point - that tells you the denominator as a power of 10, making the conversion systematic.

Question 3. Write each of the following in standard form.
(i) The height of the Mount Everest is 8848 m.
(ii) The speed of light is 300000000 m/s.
(iii) The Sun - Earth distance is 149500000000 m.
Answer:
(i) The height of the Mount Everest is 8848 m. In standard form, we have: \( 8848 = 8.848 \times 1000 \text{ m} = 8.848 \times 10^3 \text{ m} \)
(ii) The speed of light is 300000000 m/s. In standard form, we have: \( 300000000 = 3 \times 100000000 \text{ m/s} = 3 \times 10^8 \text{ m/s} \)
(iii) The Sun - Earth distance is 149500000000 m. In standard form, we have: \( 149500000000 = 1496 \times 100000000 = 1.496 \times 1000 \times 100000000 = 1.496 \times 10^3 \times 10^8 = 1.496 \times 10^{11} \text{ m} \)
In simple words: Place the decimal point after the first non-zero digit. Count the number of places from the original decimal point to the new position - that is your power of 10.

Exam Tip: For very large numbers, break them into groups of three digits from right to left (like commas) - this helps you count the exponent accurately without losing track.

Question 4. Simplify the following expressions and express the results in standard form.
Answer: The working shown uses exponent laws to combine and simplify each expression, rewriting them with a coefficient between 1 and 10 multiplied by the appropriate power of 10. This is the definition of standard form.
In simple words: Apply the exponent rules you know to combine like terms and powers of 10, then rearrange so your final answer has exactly one digit before the decimal point.

Exam Tip: Always verify that your final coefficient is between 1 and 10 (not 0.5 or 15) - a quick glance confirms you're in proper standard form.

Question 5. Express each of the following in standard form.
(i) 0.0006
(ii) 0.00000083
(iii) 0.00000534
(iv) 0.0027
(v) 0.00000165
(vi) 0.00000000689
Answer:
(i) \( 0.0006 = \frac{6}{10^4} = 6 \times 10^{-4} \)
(ii) \( 0.00000083 = \frac{83}{10^8} = 8.3 \times 10^{(1-8)} = 8.3 \times 10^{-7} \)
(iii) \( 0.00000534 = \frac{534}{10^8} = 5.34 \times 10^{(2-10)} = 5.34 \times 10^{-8} \)
(iv) \( 0.0027 = \frac{27}{10^4} = 2.7 \times 10^{(1-4)} = 2.7 \times 10^{-3} \)
(v) \( 0.00000165 = \frac{165}{10^8} = 1.65 \times 10^{(2-8)} = 1.65 \times 10^{-6} \)
(vi) \( 0.00000000689 = \frac{689}{10^{11}} = 6.89 \times 10^{(2-11)} = 6.89 \times 10^{-9} \)
In simple words: For decimals less than 1, count how many places to move the decimal point to the right to get a number between 1 and 10. That count becomes a negative exponent on 10.

Exam Tip: Count the zeros after the decimal point before the first non-zero digit - that's usually one less than your final exponent's absolute value.

Question 6. Write each of the following in standard form.
(i) 1 micron
(ii) 0.0000004 m
(iii) Thickness of paper = 0.03 mm
Answer:
(i) 1 micron \( = \frac{1}{1000000} \text{ m} = 1 \times 10^{-6} \text{ m} \)
(ii) \( 0.0000004 \text{ m} = \frac{4}{10^7} \text{ m} = \left(4 \times 10^{-7}\right) \text{ m} \)
(iii) Thickness of paper \( = 0.03 \text{ mm} = \frac{3}{10^2} \text{ mm} = \left(3 \times 10^{-2}\right) \text{ mm} \)
In simple words: Identify the first significant (non-zero) digit and position your decimal after it. Count the movement and use a negative power of 10 for decimals smaller than 1.

Exam Tip: For measurements of tiny objects (microns, millimeters), standard form with negative exponents is the clearest way to communicate the size.

Question 7. Express each of the following as a decimal.
(i) \( 2.06 \times 10^{-5} \)
(ii) \( 5 \times 10^{-7} \)
(iii) \( 6.82 \times 10^{-6} \)
(iv) \( 5.673 \times 10^{-4} \)
(v) \( 1.8 \times 10^{-2} \)
(vi) \( 4.129 \times 10^{-3} \)
Answer:
(i) \( 2.06 \times 10^{-5} = \frac{206}{100} \times \frac{1}{10^5} = \frac{206}{10^2 \times 10^5} = \frac{206}{10^{(2+5)}} = \frac{206}{10^7} = \frac{206}{10000000} = 0.0000206 \)
(ii) \( 5 \times 10^{-7} = \frac{5}{10^7} = \frac{5}{10000000} = 0.0000005 \)
(iii) \( 6.82 \times 10^{-6} = \frac{682}{100} \times \frac{1}{10^6} = \frac{682}{10^2 \times 10^6} = \frac{682}{10^{(2+6)}} = \frac{682}{10^8} = \frac{682}{100000000} = 0.00000682 \)
(iv) \( 5.673 \times 10^{-4} = \frac{5673}{1000} \times \frac{1}{10^4} = \frac{5673}{10^3 \times 10^4} = \frac{5673}{10^{(3+4)}} = \frac{5673}{10^7} = \frac{5673}{10000000} = 0.0005673 \)
(v) \( 1.8 \times 10^{-2} = \frac{18}{10} \times \frac{1}{10^2} = \frac{18}{10 \times 10^2} = \frac{18}{10^3} = \frac{18}{1000} = 0.018 \)
(vi) \( 4.129 \times 10^{-3} = \frac{4129}{1000} \times \frac{1}{10^3} = \frac{4129}{10^3 \times 10^3} = \frac{4129}{10^{(3+3)}} = \frac{4129}{10^6} = \frac{4129}{1000000} = 0.004129 \)
In simple words: A negative exponent means the decimal goes to the left. The magnitude of the exponent tells you how many zeros follow the decimal point before the significant digits.

Exam Tip: Write out the fraction form first, then count decimal places - this prevents mistakes with where the decimal ends up.

Exercise 2C

Question 1. Simplify the following.
Answer: (c) \( \frac{125}{8} \)

\( \left(\frac{2}{5}\right)^{-3} = \left(\frac{5}{2}\right)^3 = \frac{5^3}{2^3} = \frac{125}{8} \)
In simple words: Flip the fraction upside down when the exponent is negative. Then raise both numerator and denominator to that power separately.

Exam Tip: Always flip before raising to a power - this is the correct order and prevents calculation errors.

Question 2. Simplify the following.
Answer: (d) \( \frac{1}{81} \)

\( (-3)^{-4} = \frac{1}{(-3)^4} = \frac{1}{(-1)^4 \times 3^4} = \frac{1}{(3)^4} = \frac{1}{81} \)
In simple words: A negative exponent means "take the reciprocal". An even exponent on a negative number produces a positive result, so the answer is positive.

Exam Tip: Pay attention to whether the exponent is even or odd - this determines the sign of your final answer.

Question 3. Simplify the following.
Answer: (b) \( -\frac{1}{32} \)

\( (-2)^{-5} = \frac{1}{(-2)^5} = \frac{1}{-32} = -\frac{1 \times (-1)}{32 \times (-1)} = -\frac{1}{32} \)
In simple words: Apply the negative exponent rule to get the reciprocal. An odd exponent on a negative number stays negative.

Exam Tip: Odd exponents preserve the sign, even exponents remove it - always check this before finalizing your answer.

Question 4. Simplify the following.
Answer: (d) \( \frac{1}{8} \)

\( \left(2^{-5} \div 2^{-4}\right) = \left(\frac{1}{2^5} \div \frac{1}{2^4}\right) = \left(\frac{1}{32} \div \frac{1}{16}\right) = \left(\frac{1}{32} \times 4\right) = \frac{4}{32} = \frac{1}{8} \)
In simple words: When dividing by a fraction, multiply by its reciprocal instead. Simplify the result by finding the greatest common divisor.

Exam Tip: Use the rule a^m ÷ a^n = a^(m-n) - this is faster than converting to fractions first.

Question 5. Simplify the following.
Answer: (b) \( \frac{60}{7} \)

\( \left(3^{-1} + 4^{-1}\right)^{-1} \div 5^{-1} = \left(\frac{1}{3} + \frac{1}{4}\right)^{-1} \div \frac{1}{5} = \left(\frac{4 + 3}{12}\right)^{-1} \div \frac{1}{5} = \left(\frac{7}{12}\right)^{-1} \div \frac{1}{5} = \frac{12}{7} \div \frac{1}{5} = \frac{12}{7} \times 5 = \frac{60}{7} \)
In simple words: Work inside the parentheses first. Add the fractions by finding a common denominator. Then flip to handle the negative exponent. Finally, divide by using multiplication of the reciprocal.

Exam Tip: Always combine fractions inside brackets before applying the outer exponent - this greatly simplifies the work.

Question 6. Simplify the following.
Answer: (c) 29

\( \left(\frac{1}{2}\right)^{-2} + \left(\frac{1}{3}\right)^{-2} + \left(\frac{1}{4}\right)^{-2} = 2^2 + 3^2 + 4^2 = 4 + 9 + 16 = 29 \)
In simple words: A negative exponent on a fraction flips the fraction and removes the minus sign from the exponent. So each term becomes a simple integer squared.

Exam Tip: Recognize that \( \left(\frac{1}{a}\right)^{-n} = a^n \) - memorizing this pattern saves time on calculations.

Question 7. Simplify the following.
Answer: (a) \( \frac{19}{64} \)

\( \left[\left(\frac{1}{3}\right)^{-3} - \left(\frac{1}{2}\right)^{-3}\right] \div \left(\frac{1}{4}\right)^{-3} = \left[3^3 - 2^3\right] \div 4^3 = [27 - 8] \div 64 = 19 \div 64 = \frac{19}{64} \)
In simple words: Handle each negative exponent separately inside the brackets by flipping the fraction base. Calculate the subtraction. Then divide by the last term.

Exam Tip: Do negative exponents first, before any subtraction or division - this keeps the problem clear and organized.

Question 8. Simplify the following.
Answer: (a) \( \frac{1}{16} \)

\( \left[\left[\left(-\frac{1}{2}\right)^{2}\right]^{-2}\right]^{-1} = \left[\left[\left(-\frac{1}{2}\right)^{2}\right]^{-2}\right]^{-1} = \left[\left\{-\frac{1}{2}\right\}^{-4}\right]^{-1} = \left(-\frac{1}{2}\right)^{(-4 \times -1)} = \left(-\frac{1}{2}\right)^{4} = \frac{1}{16} \)
In simple words: Apply the power-of-a-power rule by multiplying exponents. Work from the inside brackets outward, simplifying one layer at a time.

Exam Tip: Multiply exponents carefully, watching for sign changes - a double negative becomes positive.

Question 9. Find x: \( \left(\frac{1}{12}\right)^{4} \times \left(\frac{1}{12}\right)^{3x} = \left(\frac{1}{12}\right)^{5} \).
Answer: (d) 3

\( \left(\frac{1}{12}\right)^{4 + 3x} = \left(\frac{1}{12}\right)^{5} \)

\( \Rightarrow 4 + 3x = 5 \)

\( \Rightarrow 3x = 5 - 4 = 1 \) 

\( \Rightarrow 3x - 4 = 5 \)

\( \Rightarrow 3x = 9 \)

\( \text{or } x = \frac{9}{3} = 3 \)
In simple words: Combine the exponents on the left by adding them. Match the bases - since they're equal, the exponents must be equal too. Solve the resulting simple equation for x.

Exam Tip: When bases are identical on both sides, you can ignore them and focus only on equating the exponents - this is a huge shortcut.

Question 10. Solve: \( \left(2^{3x-1} + 10\right) \div 7 = 6 \).
Answer: (d) 2

\( \left(2^{3x-1} + 10\right) \div 7 = 6 \)

\( \Rightarrow \frac{(2^{3x-1} + 10)}{7} = 6 \)

\( \text{On cross multiplying:} \)

\( \left(2^{3x-1} + 10\right) \times 1 = 6 \times 7 = 42 \)

\( \Rightarrow 2^{3x-1} = 42 - 10 \)

\( \Rightarrow 2^{3x-1} = 32 \)

\( \Rightarrow 2^{3x-1} = 2^5 \)

\( \Rightarrow 3x - 1 = 5 \)

\( \Rightarrow 3x = 6 \)

\( \text{Therefore, } x = 2 \)
In simple words: Isolate the exponential term by moving constants. Convert the resulting number to the same base as the left side. Equate exponents and solve.

Exam Tip: Always check if the final number can be expressed as a power of the base - if yes, you can equate exponents directly for a clean solution.

Question 11. Show that \( \left(\frac{3}{5}\right)^{0} = 1 \) using the law of exponents.
Answer: (c) 1

\( \text{Using the law of exponents } \left(\frac{a}{b}\right)^{0} = 1: \)

\( \therefore \left(\frac{3}{5}\right)^{0} = 1 \)
In simple words: Any non-zero number or fraction raised to the power zero always equals 1. This is a fundamental rule of exponents.

Exam Tip: This zero exponent rule is one of the most useful and frequently tested - memorize it and spot it quickly in problems.

Question 12. Simplify the following.
Answer: (c) \( -\frac{3}{5} \)

\( \left(\frac{-5}{3}\right)^{-1} = \left(\frac{-1}{3}\right)^{-1} = -\frac{3}{(- 5)(-1)} = -\frac{3}{5(-1)} = -\frac{3}{-5} = -\frac{3}{5} \)
In simple words: Flip the fraction upside down to apply the negative exponent. Keep track of the negative sign throughout the operation.

Exam Tip: Negative exponent of -1 means taking the reciprocal - so just flip the fraction and keep the same sign structure.

Question 13. Simplify the following.
Answer: (d) \( -\frac{1}{8} \)

\( \left(\frac{-1}{2}\right)^{3} = \frac{(-1)^3}{2^3} = \frac{-1}{8} \)
In simple words: Raise both the numerator and denominator to the given power. An odd exponent on a negative keeps it negative.

Exam Tip: Separate the numerator and denominator, apply the exponent to each, then simplify - this method never fails.

Question 14. Simplify the following.
Answer: (b) \( \frac{9}{16} \)

\( \left(\frac{-3}{4}\right)^{2} = \frac{(-3)^2}{(4)^2} = \frac{9}{16} \)
In simple words: Square both top and bottom separately. An even exponent turns any negative base positive.

Exam Tip: Remember that even exponents make everything positive - this is the key insight for problems with negative bases.

Question 15. Express in standard form: 3670000.
Answer: (c) \( 3.67 \times 10^6 \)

\( 3670000 = 367 \times 10^4 = 3.67 \times 100 \times 10^4 = 3.67 \times 10^2 \times 10^4 = 3.67 \times 10^{(2+4)} = 3.67 \times 10^6 \)
In simple words: Move the decimal point between the first and second digits. Count how many places you moved - that count is your exponent on 10.

Exam Tip: For large numbers, always check that the coefficient is between 1 and 10 before confirming your answer is in standard form.

Question 16. Express in standard form: 0.000463.
Answer: (b) \( 4.63 \times 10^{-5} \)

\( 0.000463 = \frac{463}{10^6} = \frac{4.63 \times 10^2}{10^7} = 4.63 \times 10^{(2-7)} = 4.63 \times 10^{-5} \)
In simple words: For decimals, move the decimal point to the right of the first non-zero digit. Count how many places right you moved - that count becomes your negative exponent on 10.

Exam Tip: The number of zeros after the decimal is usually one more than the absolute value of your exponent - this quick check verifies the exponent.

Question 17. Express as a decimal: 0.000367 × 10⁴.
Answer: (a) 3.67

\( 0.000367 \times 10^4 = \frac{367}{10^6} \times 10^4 = 367 \times 10^{(4-6)} = 367 \times 10^{-2} = \frac{367}{10^2} = \frac{367}{100} = 3.67 \)
In simple words: Multiply the decimal by the power of 10. This shifts the decimal point to the right by 4 places, yielding a simple decimal result.

Exam Tip: When multiplying by a positive power of 10, move the decimal right; for negative powers, move left - count the exponent to know how far.