To solve the equation [tex]\((x-5)^{\frac{3}{4}} + 10 = 12\)[/tex], we will isolate [tex]\(x\)[/tex]. Let's follow the steps systematically:
1. Isolate the term with [tex]\(x\)[/tex]:
[tex]\[
(x-5)^{\frac{3}{4}} + 10 = 12
\][/tex]
Subtract 10 from both sides:
[tex]\[
(x-5)^{\frac{3}{4}} = 2
\][/tex]
2. Eliminate the exponent [tex]\(\frac{3}{4}\)[/tex]:
To do this, raise both sides to the power of [tex]\(\frac{4}{3}\)[/tex], which is the reciprocal of [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
\left( (x-5)^{\frac{3}{4}} \right)^{\frac{4}{3}} = 2^{\frac{4}{3}}
\][/tex]
Simplifying the left side, we get:
[tex]\[
x - 5 = 2^{\frac{4}{3}}
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Add 5 to both sides:
[tex]\[
x = 2^{\frac{4}{3}} + 5
\][/tex]
4. Simplify the right side:
Let's interpret [tex]\(2^{\frac{4}{3}}\)[/tex]:
[tex]\[
2^{\frac{4}{3}} = \left( 2^4 \right)^{\frac{1}{3}} = 16^{\frac{1}{3}}
\][/tex]
[tex]\(16\)[/tex] can be rewritten as [tex]\(2^4\)[/tex], so:
[tex]\[
16^{\frac{1}{3}} = (2^4)^{\frac{1}{3}} = 2^{\frac{4}{3}}
\][/tex]
Therefore:
[tex]\[
2^{\frac{4}{3}} = \sqrt[3]{16}
\][/tex]
5. Combine the terms:
[tex]\[
x = \sqrt[3]{16} + 5
\][/tex]
Thus, the solution to the equation [tex]\((x-5)^{\frac{3}{4}} + 10 = 12\)[/tex] is:
[tex]\[
x = \sqrt[3]{16} + 5
\][/tex]
