How do you solve a negative exponent?

To solve a negative exponent, take the reciprocal of the base and make the exponent positive. For example, a to the power of -n becomes 1 divided by a to the power of n.

What happens when a fraction has a negative exponent?

When a fraction is raised to a negative exponent, you flip the fraction (take its reciprocal) and change the exponent to a positive number. For example, (a/b) to the power of -n becomes (b/a) to the power of n.

Is zero with a negative exponent defined?

No, zero with a negative exponent is undefined. Applying the negative exponent rule would require dividing by zero, which is not mathematically possible.

Negative Exponents: Questions and Step-by-Step Solutions

This page is designed to help students, parents, and teachers master the topic of negative exponents through carefully selected questions and step-by-step solutions with explanations. Expand the hidden solutions beneath each question to reveal the core rules and reasoning behind every step.

The questions on this page cover essential algebraic skills, including:

Applying the negative exponent rule to whole numbers and fractions
Evaluating expressions that combine multiplication, addition, and negative powers
Understanding the order of operations when dealing with negative bases vs. negative signs outside the base
Identifying undefined expressions such as division by zero

Rule Name	Formula
Negative Exponent Rule	\( a^{-n} = \dfrac{1}{a^n} \)
Negative Exponent on a Fraction	\( \left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^n \)
Power of a Product	\( (a \times b)^n = a^n \times b^n \)
Power of a Quotient	\( \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n} \)
Zero Exponent Rule	\( a^0 = 1 \) (where \( a \neq 0 \))

Practice Questions & Solutions

Simplify the expression: \( 5^{-2} \)

View Step-by-Step Solution

Step 1: Apply the negative exponent rule \( a^{-n} = \dfrac{1}{a^n} \).

\[ 5^{-2} = \dfrac{1}{5^2} \]

Step 2: Evaluate the square in the denominator.

\[ = \dfrac{1}{25} \]
Evaluate: \( \left( -\dfrac{1}{3} \right)^{-2} \)

View Step-by-Step Solution

Step 1: Use the fraction negative exponent rule \( \left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^n \). Flip the fraction and make the exponent positive.

\[ \left(-\dfrac{1}{3}\right)^{-2} = \left(-\dfrac{3}{1}\right)^2 \]

Step 2: Simplify the fraction inside the parentheses.

\[ = (-3)^2 \]

Step 3: A negative number squared becomes positive.

\[ = 9 \]
Simplify the expression: \( \dfrac{5^{-1}}{3^{-1}} \)

View Step-by-Step Solution

Step 1: Apply the negative exponent rule to move factors across the fraction bar. A negative exponent in the numerator moves to the denominator, and a negative exponent in the denominator moves to the numerator.

\[ \dfrac{5^{-1}}{3^{-1}} = \dfrac{3^1}{5^1} \]

Step 2: Simplify.

\[ = \dfrac{3}{5} \]
Evaluate: \( (2^{-3}) \times (3^{-2}) \)

View Step-by-Step Solution

Step 1: Rewrite each term using the negative exponent rule.

\[ (2^{-3})(3^{-2}) = \dfrac{1}{2^3} \times \dfrac{1}{3^2} \]

Step 2: Evaluate the exponents.

\[ = \dfrac{1}{8} \times \dfrac{1}{9} \]

Step 3: Multiply the fractions.

\[ = \dfrac{1}{72} \]
Simplify the expression: \( -2^{-3} \)

View Step-by-Step Solution

Step 1: Understand the order of operations. The exponent applies only to the base \(2\), not to the negative sign.

\[ -2^{-3} = -\left(2^{-3}\right) \]

Step 2: Apply the negative exponent rule to the base.

\[ = -\left(\dfrac{1}{2^3}\right) \]

Step 3: Evaluate the exponent.

\[ = -\dfrac{1}{8} \]
Evaluate: \( -1^{-3} + 2^{-3} \)

View Step-by-Step Solution

Step 1: Apply exponents before addition or negation. Note that in \( -1^{-3} \), the exponent applies only to the \(1\).

\[ -\left(\dfrac{1}{1^3}\right) + \dfrac{1}{2^3} \]

Step 2: Evaluate the denominators.

\[ = -1 + \dfrac{1}{8} \]

Step 3: Find a common denominator to add the terms.

\[ = -\dfrac{8}{8} + \dfrac{1}{8} = -\dfrac{7}{8} \]
Evaluate: \( (-1)^{-4} + 2^{-3} \)

View Step-by-Step Solution

Step 1: Here, the parentheses mean the negative sign is part of the base. Apply the negative exponent rule.

\[ \dfrac{1}{(-1)^4} + \dfrac{1}{2^3} \]

Step 2: Evaluate the exponents. A negative number to an even power is positive.

\[ = \dfrac{1}{1} + \dfrac{1}{8} = 1 + \dfrac{1}{8} \]

Step 3: Find a common denominator to add.

\[ = \dfrac{8}{8} + \dfrac{1}{8} = \dfrac{9}{8} \]
Simplify the expression: \( \left( -4^{-2} \right) \left( 2^{2} \right) \)

View Step-by-Step Solution

Step 1: Address the first term. The exponent \(-2\) applies only to the \(4\), not the negative sign.

\[ \left(-\dfrac{1}{4^2}\right) \times (2^2) \]

Step 2: Evaluate the exponents.

\[ = \left(-\dfrac{1}{16}\right) \times 4 \]

Step 3: Multiply and simplify the fraction.

\[ = -\dfrac{4}{16} = -\dfrac{1}{4} \]
Evaluate: \( (-1)^{-3} + 2^{0} \)

View Step-by-Step Solution

Step 1: Apply the negative exponent rule to the first term, and remember the Zero Exponent Rule (\(a^0 = 1\)) for the second term.

\[ \dfrac{1}{(-1)^3} + 1 \]

Step 2: Evaluate the exponent. A negative number to an odd power remains negative.

\[ = \dfrac{1}{-1} + 1 \]

Step 3: Simplify.

\[ = -1 + 1 = 0 \]
Evaluate: \( 0^{-3} \)

View Step-by-Step Solution

Step 1: Rewrite using the negative exponent rule.

\[ 0^{-3} = \dfrac{1}{0^3} \]

Step 2: Evaluate the denominator.

\[ = \dfrac{1}{0} \]

Step 3: Division by zero is mathematically impossible. Therefore, the expression is undefined.

Negative Exponents: Questions and Step-by-Step Solutions - Grade 8

Review: Core Rules of Exponents

Practice Questions & Solutions

Links and References