Simplify radicals and exponents sqrt[3]{(x^{3} y)^{2}y^{4}}

Question
Simplify radicals and exponents \(\sqrt[3]{(x^{3}\ y)^{2}y^{4}}\)

Answers (1)

2021-03-02
Distribute the exponent 2 inside the bracket.
Formula: \((a^{b})^{c} = a^{bc}\)
\(\sqrt[3]{(x^{3}\ y)^{2}y^{4}}\)
\(=\sqrt[3]{x^{3\ \times\ 2}y^{2}y^{4}}\)
\(=\sqrt[3]{x^{6}y^{2}y^{4}}\)
Since the bases are for y, so add the exponents.
Formula: \(a^{b}\ \cdot\ a^{c} = a^{b + c}\)
\(=\sqrt[3]{x^{6}y^{2}y^{4}}\)
\(=\sqrt[3]{x^{6}y^{2\ +\ 4}}\)
\(=\sqrt[3]{x^{6}y^{6}}\)
Because of the cube root we have to divide the exponents by 3.
\(\sqrt[3]{x^{6}y^{6}}\)
\(= x^{\frac{6}{3}}y^{\frac{6}{3}}\)
\(= x^{2} y^{2}\)
Finally answer: \(x^{2} y^{2}\)
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