Questions: IXC My IXL Learning Assessment Analytics IXL. Spotlight - Algebra 1> EE, 10 Checkpoint: Radicals and rational exponents KTK Which of the following are equivalent to 1/4 ? Select all that apply. (8^-3)^(1/3) (sqrt[3]64)^2 sqrt[4]1/16 (sqrt[3]1/8)^2

Evaluate \(\left(8^{-3}\right)^{\frac{1}{3}}\)

Simplify the exponent

\(\left(8^{-3}\right)^{\frac{1}{3}} = 8^{-3 \cdot \frac{1}{3}} = 8^{-1}\)

Evaluate \(8^{-1}\)

\(8^{-1} = \frac{1}{8}\)

\(\boxed{\left(8^{-3}\right)^{\frac{1}{3}} = \frac{1}{8}}\)

Evaluate \((\sqrt[3]{64})^{2}\)

Simplify \(\sqrt[3]{64}\)

\(\sqrt[3]{64} = 4\)

Square the result

\(4^{2} = 16\)

\(\boxed{(\sqrt[3]{64})^{2} = 16}\)

Evaluate \(\sqrt[4]{\frac{1}{16}}\)

Simplify the fourth root

\(\sqrt[4]{\frac{1}{16}} = \left(\frac{1}{16}\right)^{\frac{1}{4}}\)

Express \(\frac{1}{16}\) as a power of 2

\(\frac{1}{16} = 2^{-4}\), so \(\left(2^{-4}\right)^{\frac{1}{4}} = 2^{-1} = \frac{1}{2}\)

\(\boxed{\sqrt[4]{\frac{1}{16}} = \frac{1}{2}}\)

Evaluate \(\left(\sqrt[3]{\frac{1}{8}}\right)^{2}\)

Simplify \(\sqrt[3]{\frac{1}{8}}\)

\(\sqrt[3]{\frac{1}{8}} = \frac{1}{2}\)

Square the result

\(\left(\frac{1}{2}\right)^{2} = \frac{1}{4}\)

\(\boxed{\left(\sqrt[3]{\frac{1}{8}}\right)^{2} = \frac{1}{4}}\)

\(\boxed{\left(8^{-3}\right)^{\frac{1}{3}} = \frac{1}{8}}\)
\(\boxed{(\sqrt[3]{64})^{2} = 16}\)
\(\boxed{\sqrt[4]{\frac{1}{16}} = \frac{1}{2}}\)
\(\boxed{\left(\sqrt[3]{\frac{1}{8}}\right)^{2} = \frac{1}{4}}\)

The expressions equivalent to \(\frac{1}{4}\) are:
\(\boxed{\left(\sqrt[3]{\frac{1}{8}}\right)^{2}}\).