Sure, let's solve the equation step-by-step:
The given equation is:
[tex]\[ 5 \cdot x^{\frac{4}{3}} = 80 \][/tex]
1. Isolate the term with the variable:
First, divide both sides of the equation by 5 to isolate [tex]\( x^{\frac{4}{3}} \)[/tex].
[tex]\[ x^{\frac{4}{3}} = \frac{80}{5} \][/tex]
Simplify the right-hand side:
[tex]\[ x^{\frac{4}{3}} = 16 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
To solve for [tex]\( x \)[/tex], we need to undo the exponent [tex]\( \frac{4}{3} \)[/tex]. We can do this by raising both sides of the equation to the reciprocal of [tex]\( \frac{4}{3} \)[/tex], which is [tex]\( \frac{3}{4} \)[/tex].
[tex]\[ x = 16^{\frac{3}{4}} \][/tex]
3. Calculate the value:
Finding [tex]\( 16^{\frac{3}{4}} \)[/tex]:
- First, understand that [tex]\( \frac{3}{4} \)[/tex] is the exponent that can be applied as:
- The fourth root of 16, raised to the power of 3, or
- The cube of the fourth root of 16.
- We know that the fourth root of 16 is 2, because:
[tex]\[ \sqrt[4]{16} = 2 \][/tex]
- Now, raise this result to the power of 3:
[tex]\[ 2^3 = 8 \][/tex]
The final value of [tex]\( x \)[/tex] is:
[tex]\[ x = 8 \][/tex]
Therefore, the correct solution is [tex]\( 8 \)[/tex].
