To solve the equation [tex]\(\frac{81}{16} = r^4\)[/tex], we need to determine the value of [tex]\(r\)[/tex].
1. Identify the given equation:
[tex]\[
\frac{81}{16} = r^4
\][/tex]
2. Recognize that we need to find [tex]\(r\)[/tex] such that [tex]\(r^4\)[/tex] equals [tex]\(\frac{81}{16}\)[/tex].
3. To isolate [tex]\(r\)[/tex], take the fourth root of both sides of the equation.
The fourth root of [tex]\(\frac{81}{16}\)[/tex] can be expressed as:
[tex]\[
r = \sqrt[4]{\frac{81}{16}}
\][/tex]
4. Calculate the fourth root of the fraction [tex]\(\frac{81}{16}\)[/tex]:
Since taking the fourth root undoes raising a number to the fourth power, we find:
[tex]\[
r = \left(\frac{81}{16}\right)^{\frac{1}{4}}
\][/tex]
5. Evaluate the expression:
[tex]\(\left(\frac{81}{16}\right)^{1/4} = 1.5\)[/tex]
Hence,
[tex]\[
r = 1.5
\][/tex]
6. Verify the solution:
To ensure our solution is correct, we can check if [tex]\((1.5)^4\)[/tex] equals [tex]\(\frac{81}{16}\)[/tex]:
[tex]\[
1.5^4 = (1.5 \times 1.5 \times 1.5 \times 1.5) = 5.0625
\][/tex]
Confirming the calculation, we see that:
[tex]\[
\left(\frac{81}{16}\right) = 5.0625
\][/tex]
So, the value [tex]\(r\)[/tex] that satisfies the equation [tex]\(\frac{81}{16} = r^4\)[/tex] is [tex]\(r = 1.5\)[/tex].
Therefore, our solution is accurately found as:
[tex]\[
r = \boxed{1.5}
\][/tex]
