Exponent Rules: Power of a Power

Overview

The Power of a Power rule states that when raising an exponential expression to another power, you multiply the exponents. In mathematical terms:

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

This rule applies to simplify expressions where an exponent is raised to another exponent.

Steps to Apply the Power of a Power Rule

  1. Identify the base and the exponents.
  2. Multiply the exponents together.
  3. Write the base with the new exponent.

Example 1: Simplifying with Numbers

Consider the following expression:

\((3^4)^2\)

Step 1: Multiply the exponents 4 and 2:

\(3^{(4 \cdot 2)} = 3^8\)

Thus, the simplified expression is:

\(3^8\)

Example 2: Simplifying with Variables

Consider the following expression:

\((x^5)^3\)

Step 1: Multiply the exponents 5 and 3:

\(x^{(5 \cdot 3)} = x^{15}\)

Thus, the simplified expression is:

\(x^{15}\)

Practice Questions

  1. Question 1: Simplify the following expression:

    \((2^3)^4\)

    Solution

    Step 1: Multiply the exponents 3 and 4:

    \(2^{(3 \cdot 4)} = 2^{12}\)

  2. Question 2: Simplify the following expression:

    \((y^7)^2\)

    Solution

    Step 1: Multiply the exponents 7 and 2:

    \(y^{(7 \cdot 2)} = y^{14}\)

  3. Question 3: Simplify the following expression:

    \((5^2)^3\)

    Solution

    Step 1: Multiply the exponents 2 and 3:

    \(5^{(2 \cdot 3)} = 5^6\)

  4. Question 4: Simplify the following expression:

    \((a^{10})^4\)

    Solution

    Step 1: Multiply the exponents 10 and 4:

    \(a^{(10 \cdot 4)} = a^{40}\)

  5. Question 5: Simplify the following expression:

    \((3^5)^2\)

    Solution

    Step 1: Multiply the exponents 5 and 2:

    \(3^{(5 \cdot 2)} = 3^{10}\)

→Exponent Rules: Power of a Product