Subjects

Arithmetic

Exponents

Working with Powers of 10

Arithmetic

Dec 5, 2025

2 pages

Understanding the Power Rule in Mathematics

Chelsea @zuncho

The power rule is a key concept in algebra that shows you how to simplify expressions with powers... Show more

Name:
Date: Jan 30 Hour: 5th
Unit 5 Day 7: The Power Rule
Focus Question: How do I simplify a power to a power?

A. Mary says she can expand

The Power Rule Basics

Ever wondered what happens when you raise a power to another power? That's exactly what the power rule helps us solve! When you have $(2^3)^2$ , you're actually multiplying $2^3$ by itself twice $(2^3) \cdot (2^3)$ . This expands to $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ , which equals $2^6$ .

Looking at examples like $(5^3)^4$ and $(x^2)^5$ , a pattern emerges. In each case, when we expand and count all the factors, we find that the exponents multiply together. This gives us the power rule $(a^m)^n = a^{m \cdot n}$ . So for example, $(5^3)^4 = 5^{3 \cdot 4} = 5^{12}$ .

The power rule works with negative exponents too. For instance, $(x^4)^{-2} = x^{4 \cdot (-2)} = x^{-8}$ , which can also be written as $\frac{1}{x^8}$ . When both exponents are negative, like in $(k^{-8})^{-3}$ , they multiply to give a positive result $k^{(-8) \cdot (-3)} = k^{24}$ .

💡 If you ever forget the power rule, you can always go back to basics! Just write out the expression in expanded form and count how many times the base appears.

Power Rule with Multiple Bases

The power rule gets even more useful when dealing with expressions that have more than one base. For example, when simplifying $(4x)^2$ , we're actually calculating $(4x) \cdot (4x)$ , which equals $4 \cdot 4 \cdot x \cdot x$ or $4^2 \cdot x^2 = 16x^2$ .

This shows us another important rule when raising a product to a power, you can distribute the exponent to each factor. Mathematically, this is written as (ab)^m = a^m · b^m. For example, $(xy)^4 = x^4y^4$ and $(12m)^3 = 12^3m^3 = 1728m^3$ .

When solving mixed practice problems, combine these rules carefully. For expressions like $(5h^7)^3$ , apply the distribution rule first $(5h^7)^3 = 5^3 \cdot (h^7)^3 = 125 \cdot h^{21}= 125h^{21}$ . For problems with fractions like $(\frac{a^2b}{a^4})^3$ , distribute the exponent to both numerator and denominator, then apply the power rule to each term.

🔑 Remember When simplifying complex exponential expressions, work step by step! First distribute exponents across products, then apply the power rule to each term, and finally combine like terms.

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Subjects

Arithmetic

Exponents

Working with Powers of 10

Arithmetic

•

Dec 5, 2025

•

2 pages

Understanding the Power Rule in Mathematics

Chelsea

@zuncho

The power rule is a key concept in algebra that shows you how to simplify expressions with powers raised to powers. Understanding this rule helps you solve complex exponential problems quickly without having to expand everything the long way.

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The Power Rule Basics

Ever wondered what happens when you raise a power to another power? That's exactly what the power rule helps us solve! When you have $(2^3)^2$ , you're actually multiplying $2^3$ by itself twice: $(2^3) \cdot (2^3)$ . This expands to $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ , which equals $2^6$ .

Looking at examples like $(5^3)^4$ and $(x^2)^5$ , a pattern emerges. In each case, when we expand and count all the factors, we find that the exponents multiply together. This gives us the power rule: $(a^m)^n = a^{m \cdot n}$ . So for example, $(5^3)^4 = 5^{3 \cdot 4} = 5^{12}$ .

💡 If you ever forget the power rule, you can always go back to basics! Just write out the expression in expanded form and count how many times the base appears.

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Power Rule with Multiple Bases

This shows us another important rule: when raising a product to a power, you can distribute the exponent to each factor. Mathematically, this is written as (ab)^m = a^m · b^m. For example, $(xy)^4 = x^4y^4$ and $(12m)^3 = 12^3m^3 = 1728m^3$ .

When solving mixed practice problems, combine these rules carefully. For expressions like $(5h^7)^3$ , apply the distribution rule first: $(5h^7)^3 = 5^3 \cdot (h^7)^3 = 125 \cdot h^{21}= 125h^{21}$ . For problems with fractions like $(\frac{a^2b}{a^4})^3$ , distribute the exponent to both numerator and denominator, then apply the power rule to each term.

🔑 Remember: When simplifying complex exponential expressions, work step by step! First distribute exponents across products, then apply the power rule to each term, and finally combine like terms.