To solve for [tex]\( x \)[/tex] in the equation [tex]\( x^{\frac{4}{3}} = 16 \)[/tex], follow these steps:
1. Rewrite the equation for clarity:
[tex]\[
x^{\frac{4}{3}} = 16
\][/tex]
2. Isolate [tex]\( x \)[/tex] by raising both sides of the equation to the reciprocal of [tex]\(\frac{4}{3}\)[/tex], which is [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
\left(x^{\frac{4}{3}}\right)^{\frac{3}{4}} = 16^{\frac{3}{4}}
\][/tex]
3. Simplify the left side:
[tex]\[
x^{\frac{4}{3} \cdot \frac{3}{4}} = x^1 = x
\][/tex]
4. Evaluate the right side [tex]\( 16^{\frac{3}{4}} \)[/tex]:
To do this, consider the definition of fractional exponents:
[tex]\[
16^{\frac{3}{4}} = (16^{\frac{1}{4}})^3
\][/tex]
First, find [tex]\( 16^{\frac{1}{4}} \)[/tex]:
[tex]\[
16 = 2^4 \quad \text{so} \quad 16^{\frac{1}{4}} = (2^4)^{\frac{1}{4}} = 2
\][/tex]
Now raise 2 to the power of 3:
[tex]\[
(16^{\frac{1}{4}})^3 = 2^3 = 8
\][/tex]
Thus, the solution to the equation [tex]\( x^{\frac{4}{3}} = 16 \)[/tex] is:
[tex]\[
\boxed{8}
\][/tex]
