To solve the equation [tex]\(5x^{\frac{4}{3}} = 80\)[/tex], follow these steps:
1. Isolate the term with the exponent: To make [tex]\(x^{\frac{4}{3}}\)[/tex] the subject, divide both sides of the equation by 5.
[tex]\[
5x^{\frac{4}{3}} = 80
\][/tex]
[tex]\[
x^{\frac{4}{3}} = \frac{80}{5}
\][/tex]
[tex]\[
x^{\frac{4}{3}} = 16
\][/tex]
2. Solve for [tex]\(x\)[/tex]: To isolate [tex]\(x\)[/tex], raise both sides of the equation to the reciprocal of the exponent [tex]\(\frac{4}{3}\)[/tex]. The reciprocal of [tex]\(\frac{4}{3}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
[tex]\[
\left(x^{\frac{4}{3}}\right)^{\frac{3}{4}} = 16^{\frac{3}{4}}
\][/tex]
[tex]\[
x = 16^{\frac{3}{4}}
\][/tex]
3. Simplify the expression: Evaluate [tex]\(16^{\frac{3}{4}}\)[/tex].
- Recall that [tex]\(16 = 2^4\)[/tex], so:
[tex]\[
16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}}
\][/tex]
- Using the properties of exponents, [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
(2^4)^{\frac{3}{4}} = 2^{4 \cdot \frac{3}{4}} = 2^3 = 8
\][/tex]
Therefore, the solution to the equation [tex]\(5x^{\frac{4}{3}} = 80\)[/tex] is [tex]\(x = 8\)[/tex].
