Evaluate: [8]638 and Simplify: 3/(64)

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for problems 4 and 5:

4. Evaluate: $\sqrt[8]{63^8}$

Convert the radical expression to exponential form using the rule $\sqrt[n]{a^m} = a^{\frac{m}{n}}$ .
$\sqrt[8]{63^8} = 63^{\frac{8}{8}}$
Simplify the exponent.
$63^{\frac{8}{8}} = 63^1$
Evaluate the expression.
$63^1 = 63$
The evaluated expression is:
$\boxed{63}$
5. Simplify: $\sqrt[3]{\frac{297}{64}}$
Apply the property $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ to separate the numerator and denominator.
$\sqrt[3]{\frac{297}{64}} = \frac{\sqrt[3]{297}}{\sqrt[3]{64}}$
Evaluate the cube root of the denominator.
$\sqrt[3]{64} = \sqrt[3]{4^3} = 4$
Find the prime factorization of the numerator, 297.
$297 = 3 \times 99 = 3 \times 3 \times 33 = 3 \times 3 \times 3 \times 11 = 3^3 \times 11$
Substitute the prime factorization into the numerator's cube root and simplify.
$\sqrt[3]{297} = \sqrt[3]{3^3 \times 11} = \sqrt[3]{3^3} \times \sqrt[3]{11} = 3\sqrt[3]{11}$
Combine the simplified numerator and denominator.
$\frac{3\sqrt[3]{11}}{4}$
The simplified expression is:
$\boxed{\frac{3\sqrt[3]{11}}{4}}$

Answer