Certainly! Let's break down each part of the problem step by step and resolve the equations.
1. Resolving the fourth root of 81:
[tex]\[
\sqrt[4]{81}
\][/tex]
To find the fourth root of 81, we look for a number which, when raised to the power of 4, equals 81. The correct value is 3, because [tex]\(3^4 = 81\)[/tex].
[tex]\[
\sqrt[4]{81} = 3
\][/tex]
2. Calculating [tex]\(-\sqrt{9 \times 9}\)[/tex]:
[tex]\[
-\sqrt{9 \times 9}
\][/tex]
First, calculate [tex]\(9 \times 9\)[/tex]:
[tex]\[
9 \times 9 = 81
\][/tex]
Then, find the square root of 81:
[tex]\[
\sqrt{81} = 9
\][/tex]
Finally, apply the negative sign:
[tex]\[
-\sqrt{81} = -9
\][/tex]
3. Resolving the cube root of 64:
[tex]\[
\sqrt[3]{64}
\][/tex]
To find the cube root of 64, we look for a number which, when raised to the power of 3, equals 64. The correct value is approximately 4.
So,
[tex]\[
\sqrt[3]{64} \approx 4
\][/tex]
4. Resolving the fifth root of 64:
[tex]\[
\sqrt[5]{64}
\][/tex]
To find the fifth root of 64, we look for a number which, when raised to the power of 5, equals 64. The result is approximately 2.297.
So,
[tex]\[
\sqrt[5]{64} \approx 2.297
\][/tex]
5. Calculating the square root of 125:
[tex]\[
\sqrt{125}
\][/tex]
The square root of 125 is approximately 11.180.
Thus,
[tex]\[
\sqrt{125} \approx 11.180
\][/tex]
Combining all results together, we have:
[tex]\[
\begin{aligned}
\sqrt[4]{81} & = 3, \\
-\sqrt{9 \times 9} & = -9, \\
\sqrt[3]{64} & \approx 4, \\
\sqrt[5]{64} & \approx 2.297, \\
\sqrt{125} & \approx 11.180.
\end{aligned}
\][/tex]
These are the step-by-step solutions to the given mathematical expressions.
