Write $3^8$ as a power with a base of 9. Show your work.
The objective is to express the value \(3^8\) as a power with a base of \(9\). We are working with exponentiation, which is a mathematical operation involving a base and an exponent. The base is the number being multiplied by itself, and the exponent tells us how many times to use that base in the multiplication. Our given expression is \(3^8\), where the base is \(3\) and the exponent is \(8\). We want to rewrite this in the form \(9^x\) for some value \(x\).
To express \(3^8\) with a base of \(9\), we need to relate the base \(3\) to the target base \(9\). We know that \(9 = 3^2\). This relationship allows us to rewrite the base \(3\) as \(\sqrt{9}\) or \(9^{1/2}\). Alternatively, we can use the power of a power rule for exponents, which states that \((a^m)^n = a^{m \times n}\). We want to find \(x\) such that \(9^x = 3^8\). Since \(9 = 3^2\), we can substitute this into our equation:
\((3^2)^x = 3^8\)
Using the power rule, the left side becomes:
\(3^{2x} = 3^8\)
Since the bases are the same, we can equate the exponents:
\(2x = 8\)
Solving for \(x\), we divide both sides by \(2\):
\(x = \frac{8}{2} = 4\)
Thus, \(3^8 = 9^4\).
Understanding how to change the base of an exponent is essential in fields like computer science, where binary (base 2) and hexadecimal (base 16) systems are frequently converted. For example, knowing that \(16 = 2^4\) allows programmers to quickly translate between memory addresses represented in different bases. This skill also appears in finance when dealing with compound interest, as you might need to convert an annual interest rate into an equivalent monthly rate to perform accurate calculations. Mastering these exponent rules helps in simplifying complex equations and making data manipulation more efficient.