Power Rule with Negative Exponents Calculator

Learn how to apply the power rule to negative exponents with our calculator and guide. Understand the mathematical principles behind negative exponents and how they interact with the power rule in algebra.

What is the Power Rule?

The power rule is a fundamental algebraic rule that states when you multiply two exponents with the same base, you add their exponents. Mathematically, this is expressed as:

Power Rule Formula:

a^m × aⁿ = a^m+n

This rule applies when multiplying terms with the same base. For example, if you have 2³ × 2⁴, you can combine the exponents to get 2⁷.

Negative Exponents

Negative exponents represent reciprocals of the base raised to the positive exponent. The general rule is:

Negative Exponent Rule:

a^-n = 1/aⁿ

For example, 5^-2 is equivalent to 1/5², which equals 1/25.

Negative Exponents

Negative exponents can be tricky, but they follow specific rules that make calculations easier. When you encounter a negative exponent, you can rewrite the expression as a fraction with the base in the denominator.

Remember: A negative exponent indicates the reciprocal of the base raised to the positive exponent.

Combining Power Rule with Negative Exponents

When applying the power rule to expressions with negative exponents, you need to be careful about the signs. The power rule still applies, but you must consider the negative exponents as reciprocals.

Combined Rule:

a^-m × a^-n = a^-(m+n) = 1/a^m+n

For example, 3^-2 × 3^-4 = 3^-6 = 1/3⁶.

How to Use the Calculator

Our calculator makes it easy to apply the power rule with negative exponents. Simply enter the base and exponents, and the calculator will compute the result for you.

Steps to Use the Calculator

Enter the base value in the first input field.
Enter the first exponent in the second input field.
Enter the second exponent in the third input field.
Click the "Calculate" button to see the result.
Use the "Reset" button to clear the inputs and start over.

The calculator handles both positive and negative exponents, so you can use it for any combination of exponents.

Examples

Let's look at some examples to see how the power rule with negative exponents works in practice.

Example 1: Positive Exponents

Calculate 2³ × 2⁴:

Using the power rule: 2³⁺⁴ = 2⁷ = 128.

Example 2: Negative Exponents

Calculate 5^-2 × 5^-3:

Using the combined rule: 5^-(2+3) = 5^-5 = 1/5⁵ = 1/3125.

Example 3: Mixed Exponents

Calculate 4² × 4^-3:

Using the power rule: 4^2-3 = 4^-1 = 1/4.

FAQ

What is the power rule?: The power rule states that when multiplying two exponents with the same base, you add their exponents. For example, a^m × aⁿ = a^m+n.
How do negative exponents work?: Negative exponents represent reciprocals. For example, a^-n = 1/aⁿ. When applying the power rule with negative exponents, you add the exponents and keep the negative sign.
Can the power rule be used with fractions?: Yes, the power rule can be applied to fractions as long as the bases are the same. For example, (2/3)⁴ × (2/3)⁵ = (2/3)⁹.
What happens if the exponents are the same but the bases are different?: If the bases are different, you cannot combine the exponents. For example, 2³ × 3³ cannot be simplified further.
Is there a way to simplify expressions with negative exponents?: Yes, you can rewrite negative exponents as fractions or use the power rule to combine like terms. For example, 5^-2 × 5³ = 5¹ = 5.