The repeated multiplication form shows us the meaning of a power. A power represents a product of repeated, identical factors. The exponent tells us about the number of times we need to multiply the base by itself.
a^n = a * a * ... * a * a_n
We want to write the given expression in repeated multiplication form. The expression contains a power of a power, so we will rewrite the powers into the repeated multiplication form one at a time.
( 6^4 ) ^2 ⇔ 6^4 * 6^4_2
The resulting expression contains two powers. Let's rewrite both of them.
6^4 * 6^4
⇕
6 * 6 * 6 * 6_4 * 6 * 6 * 6 * 6_4
Now, we want to write the expression as a power. The product contains only identical factors, so we can join them all into one power.
6 * 6 * 6 * 6 * 6 * 6 * 6 * 6
6 appears 8 times
The number of times the 6 appears in the product is an exponent of a power. We can finally write the power!
6^8
We can see that our result confirms the Power of a Power Property. This property states that to find the power of a power, we can multiply the exponents. This means that the result of (6^4)^2 is 6 raised to the power of 4 * 2= 8, which is 6^8.
