Solve the following equation with a rational exponent: [tex]x^{\frac{2}{3}}=64[/tex].

A. 16
B. 8
C. 512
D. 32



Answer :

To solve the equation [tex]\(x^{\frac{2}{3}} = 64\)[/tex], we need to isolate [tex]\(x\)[/tex]. Start by raising both sides of the equation to the power that will eliminate the exponent on [tex]\(x\)[/tex]. In this case, the exponent is [tex]\(\frac{2}{3}\)[/tex], so we raise both sides to the power of [tex]\(\frac{3}{2}\)[/tex].

Here are the steps:

1. Write down the original equation:
[tex]\[ x^{\frac{2}{3}} = 64 \][/tex]

2. Raise both sides of the equation to the power of [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[ \left(x^{\frac{2}{3}}\right)^{\frac{3}{2}} = 64^{\frac{3}{2}} \][/tex]

3. Simplify the left side of the equation. When you raise a power to a power, you multiply the exponents:
[tex]\[ x^{\left(\frac{2}{3} \cdot \frac{3}{2}\right)} = x^1 = x \][/tex]

4. Now, calculate the right side of the equation:
[tex]\[ 64^{\frac{3}{2}} \][/tex]

5. The operation [tex]\(64^{\frac{3}{2}}\)[/tex] can be split into two steps:
- First, find the square root of 64:
[tex]\[ \sqrt{64} = 8 \][/tex]
- Then, cube the result:
[tex]\[ 8^3 = 512 \][/tex]

So [tex]\(64^{\frac{3}{2}} = 512\)[/tex].

Therefore:
[tex]\[ x = 512 \][/tex]

The solution to the equation [tex]\(x^{\frac{2}{3}} = 64\)[/tex] is [tex]\(x = 512\)[/tex].

Thus, the correct answer is:
[tex]\[ \boxed{512} \][/tex]