Problem 123
Question
Explain the power rule for exponents. Use \(\left(3^{2}\right)^{4}\) in your explanation.
Step-by-Step Solution
Verified Answer
By using the power rule, \((3^2)^4\) simplifies to \(3^8\).
1Step 1: Understand the Power Rule
The power rule for exponents states that, for any real numbers a, m, and n, when you have an exponent raised to another exponent, the exponents multiply together. This means \((a^n)^m = a^{n*m}\).
2Step 2: Identify Values
We have the expression \((3^2)^4\). Here, a = 3, n = 2, and m = 4.
3Step 3: Apply the Power Rule
Following the power rule, we multiply the exponents together: 2 * 4 = 8
4Step 4: Final Simplification
Substitute the base 'a' (which is 3) and the result of the multiplication of the exponents into the formula leading to \(3^{8}\).
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