Alright! Let's solve the given problem step-by-step.
### Part 1: Solve for [tex]\( x^2 - 81 = 0 \)[/tex]
1. The equation given is:
[tex]\[
x^2 - 81 = 0
\][/tex]
2. To solve for [tex]\( x \)[/tex], we can add 81 to both sides:
[tex]\[
x^2 = 81
\][/tex]
3. Next, take the square root of both sides:
[tex]\[
x = \pm \sqrt{81}
\][/tex]
4. Since [tex]\( \sqrt{81} = 9 \)[/tex], we get:
[tex]\[
x = \pm 9
\][/tex]
So, the solutions for [tex]\( x \)[/tex] are [tex]\( x = 9 \)[/tex] and [tex]\( x = -9 \)[/tex].
### Part 2: Calculate [tex]\( a = \sqrt[3]{x^2} \)[/tex]
1. We need to find [tex]\( \sqrt[3]{x^2} \)[/tex]. We already know that [tex]\( x \)[/tex] can be [tex]\( 9 \)[/tex] or [tex]\( -9 \)[/tex].
2. Calculate for [tex]\( x = 9 \)[/tex]:
[tex]\[
\sqrt[3]{9^2} = \sqrt[3]{81}
\][/tex]
3. Calculate for [tex]\( x = -9 \)[/tex]:
[tex]\[
\sqrt[3]{(-9)^2} = \sqrt[3]{81}
\][/tex]
4. Therefore, in both cases, we get:
[tex]\[
a = \sqrt[3]{81}
\][/tex]
### Part 3: Calculate [tex]\( b = \sqrt[3]{81} \)[/tex]
1. The value for [tex]\( \sqrt[3]{81} \)[/tex] is a cube root of 81.
2. Calculate the cube root of 81:
- Let's express 81 as [tex]\( 3^4 \)[/tex].
- The cube root of [tex]\( 3^4 \)[/tex] is [tex]\( 3^{4/3} \)[/tex].
3. Approximating, we get:
- [tex]\( 3^{4/3} \approx 4.326 \)[/tex] (approximately).
Hence, we find:
[tex]\[
b = \sqrt[3]{81} \approx 4.326
\][/tex]
### Summary
1. The solutions for [tex]\( x \)[/tex] are:
[tex]\[
x = 9 \text{ and } x = -9
\][/tex]
2. The value of [tex]\( a \)[/tex]:
[tex]\[
a = \sqrt[3]{81} \approx 4.326
\][/tex]
3. The value of [tex]\( b \)[/tex]:
[tex]\[
b = \sqrt[3]{81} \approx 4.326
\][/tex]
So, the final results are:
[tex]\[
a = \sqrt[3]{81} \approx 4.326
\][/tex]
[tex]\[
b = \sqrt[3]{81} \approx 4.326
\][/tex]
