Negative Exponent Questions: Answers and Explanations

Negative Exponents Questions with Answers presents methods and worked examples to help learners understand negative exponents and solve problems with confidence. This guide reviews the key concepts of negative exponents, including the rule  $a^{ −n} = 1/a^{ n}$​ , reciprocal form, simplifying expressions and applying exponent laws. From basic exercises to application-based questions, each solution focuses on clear steps, logical reasoning and helpful shortcuts. Worked examples with brief explanations help strengthen understanding and exam preparation.

Table of Contents

• Negative Exponent Law
• Solved Questions of Negative Exponents
• Practice Questions of Negative Exponents
• Frequently Asked Questions of Negative Exponent Questions

Negative Exponent Law

 A negative exponent tells you to take the reciprocal of the base and make the exponent positive. In symbols, $a^{ −n} = 1/a^{ n}$, where a is any non-zero number and n is a positive integer. 

 $a^{-n} = 1/a^{n}, 1/a^{-n} = a^{n}$. 

Solved Questions of Negative Exponents

Q1. Which of these is equal to 1/7⁴?

(a) 7⁴

(b) -7⁴

(c) 7⁻⁴

(d) -7⁻⁴

Solution: 

Answer: (c) 7⁻⁴

A negative exponent means ‘take the reciprocal.’ So: 7⁻⁴ = 1/7⁴

Therefore, 1/7⁴ = 7⁻⁴, so the correct answer is (c).

Q2.   Simplify:  $3^2 × 3^{-5}$ 

Solution: 

Using the product law, add the exponents:  $3^{2+(-5)} = 3^{-3},$ which equals 1/27. 

Q3.  $6^{-2} ÷ 6^{-4}$= ______

Solution: $6^{-2} ÷ 6^{-4} = 36$

Using the quotient law:  $6^{-2-(-4)} = 6^{-2+4} = 6^2 = 36$. 

Q4.  Assertion (A):  $(2^{3})^{-2} = 2^{-6}$.

Reason (R): When a power is raised to another power, the exponents are multiplied.

(a) Both A and R are true, and R correctly explains A.

(b) Both A and R are true, but R does not explain A.

(c) A is true, R is false.

(d) A is false, R is true.

Solution:

Answer: (a)Both A and R are true, and R correctly explains A.

By the power-of-a-power law,  $(2^{3})^{-2} = 2^{3×(-2)} = 2^{-6}$. The reason correctly explains why the assertion is true.

Q5. Simplify and write as a power of 2:  $(4^{-10})^{1/2}$

Solution:

First simplify the exponent:  $4^{-10 × 1/2} = 4^{-5}$. Now rewrite 4 as  $2^2$:  $(2^{2})^{-5} = 2^{-10}$.

Q6. Write $3x^{-a} × 2x^b$ as a single fraction in simplest form. 

Solution:

 $3x^{-a} = 3/x^a$. Multiplying:  $(3/x^a) × 2x^b = 6x^b/x^a$. 

Q7. True/False: $5^3 × 2^{-4}$can be simplified to  $10^{-1}$by combining the bases.

Solution:

The laws of exponents only combine exponents when the bases are the same. Since 5 and 2 are different bases, they cannot be merged. The correct simplification is 5^3 × 2^{-4} = 125 × 1/16 = 125/16.

Answer: False

Q8. Write 0.00056 in scientific notation.

Solution:

Step 1: Move the decimal point right until only one non-zero digit remains before it: 0.00056 → 5.6, moving 4 places.

Step 2: Since the original number is smaller than 1, the exponent of 10 is negative: 5.6 × 10^{-4}.

Answer: 5.6 × 10^{-4}

Q9. The width of a human hair is about  $8 × 10^{-5}$ m. Write this as an ordinary decimal number.

Solution:

A negative exponent of 10 means the decimal point shifts that many places to the left of 8: shifting 5 places gives 0.00008.

Answer: 0.00008 m

Q10. Find x if  $2^x = 1/32$.

Solution:

 $1/32 = 1/2^5 = 2^{-5}$. So  $2^x = 2^{-5}$ gives x = -5.

Answer: x = -5

Practice Questions of Negative Exponents

1. Simplify  $(8^{2})^{-1/3}$ and write the answer as a power of 2.

2. Evaluate  $(3/4)^{-3}$ and write it as a mixed number.

3. Evaluate:  $(2/3)^{-2}$

4.  The value of  $2^{-3}$is ________.

5. True/False: $(-3)^{-2}$ is a negative number.