Applying the Power of a Power Rule

Oct 13, 2025 by windailys.com 130 views

Introduction

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. When dealing with exponents, it's essential to understand the rules of exponentiation, including the power of a power rule. This article will explore the concept of equivalent expressions, focusing on the expression $\left(2 x^4 y\right)^3$ and its possible equivalent forms.

The Power of a Power Rule

The power of a power rule states that when a power is raised to another power, the exponents are multiplied. Mathematically, this can be represented as:

$(a^m)^n = a^{m \cdot n}$

where $a$ is the base, $m$ is the exponent, and $n$ is the power.

Applying the Power of a Power Rule

To find the equivalent expression for $\left(2 x^4 y\right)^3$ , we can apply the power of a power rule. By multiplying the exponents, we get:

$\left(2 x^4 y\right)^3 = 2^3 \cdot (x^4)^3 \cdot y^3$

Simplifying the Expression

Using the power of a power rule, we can simplify the expression further:

$2^3 = 8$

$(x^4)^3 = x^{4 \cdot 3} = x^{12}$

$y^3 = y^3$

Therefore, the equivalent expression for $\left(2 x^4 y\right)^3$ is:

$8 x^{12} y^3$

Comparing with the Options

Now that we have the equivalent expression, let's compare it with the given options:

A. $2 x^7 y^4$ B. $2 x^{12} y^3$ C. $8 x^7 y^4$ D. $8 x^{12} y^3$

Based on our calculation, the correct equivalent expression is:

$8 x^{12} y^3$

This matches option D.

Conclusion

In conclusion, the power of a power rule is a fundamental concept in mathematics that helps us simplify complex expressions. By applying this rule, we can find the equivalent expression for $\left(2 x^4 y\right)^3$ , which is $8 x^{12} y^3$ . This demonstrates the importance of understanding exponents and equivalent expressions in mathematics.

Key Takeaways

The power of a power rule states that when a power is raised to another power, the exponents are multiplied.
To find the equivalent expression for $\left(2 x^4 y\right)^3$ , we can apply the power of a power rule.
The equivalent expression is $8 x^{12} y^3$ .

Frequently Asked Questions

What is the power of a power rule? The power of a power rule states that when a power is raised to another power, the exponents are multiplied.
How do I apply the power of a power rule? To apply the power of a power rule, multiply the exponents of the base and the power.
What is the equivalent expression for $\left(2 x^4 y\right)^3$ ? The equivalent expression for $\left(2 x^4 y\right)^3$ is $8 x^{12} y^3$ .