We are asked to write a multiplication expression with a product of 5^(13). First, we should focus on the product.
5^(13)
Let's remember that powers are another way of representing repeating multiplication. The base — in this case 5 — is the multiplied factor. The exponent, 13, is the number of times the base is used as a factor. This allows us to rewrite our power as a product.
5^(13) = 5 * 5* ... * 5 *5^(13factors)
This is just one way of writing 5^(13) as a product. We will show that there are more. The Product of Powers rule tells us that if we multiply powers with the same base, we add their exponents.
4^2 * 4^5 = 4^(2+5) or 4^7
In our case, the result should be 5^(13). This means we want to find the powers of 5 that give 5^(13) when multiplied.
\begin{gathered}
5^\boxed{\phantom{2}} \cdot 5^\boxed{\phantom{1}} = 5^{13}
\end{gathered}
According to the Product of Powers rule, their exponents must add up to 13. Notice that there are many pairs of such numbers.
c|c
1+12= 13 & 5^1* 5^(12)=5^(13)
3+10= 13 & 5^3* 5^(10)=5^(13)
5+8= 13 & 5^5* 5^8=5^(13)
We see that each expression in the right column satisfies the conditions — they all result in a multiplication with a product of 5^(13). This is why there is not just one correct answer to the problem.
