Question

which expression is equivalent to (sqrt5{32x^{5}y^{10}z^{15}})?
(2y^{5}z^{10})
(2xy^{2}z^{3})
(6y^{5}z^{10})
(6xy^{2}z^{3})

Explanation:

Step1: Simplify the radical

Recall that \(\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}\) and \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\) (when \(a\geq0\)). For \(\sqrt[5]{32x^{5}y^{10}z^{15}}\), we can split the radical:
\(\sqrt[5]{32}\cdot\sqrt[5]{x^{5}}\cdot\sqrt[5]{y^{10}}\cdot\sqrt[5]{z^{15}}\)

Step2: Simplify each part

• For \(\sqrt[5]{32}\), since \(32 = 2^5\), then \(\sqrt[5]{2^5}=2\).
• For \(\sqrt[5]{x^{5}}\), using the rule \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), here \(n = 5\), \(m = 5\), so \(\sqrt[5]{x^{5}}=x^{\frac{5}{5}}=x\).
• For \(\sqrt[5]{y^{10}}\), \(n = 5\), \(m = 10\), so \(\sqrt[5]{y^{10}}=y^{\frac{10}{5}}=y^{2}\).
• For \(\sqrt[5]{z^{15}}\), \(n = 5\), \(m = 15\), so \(\sqrt[5]{z^{15}}=z^{\frac{15}{5}}=z^{3}\).

Step3: Multiply the simplified parts

Multiply the results from Step 2: \(2\times x\times y^{2}\times z^{3}=2xy^{2}z^{3}\)

Answer:

\(2xy^{2}z^{3}\) (corresponding to the option "2\(xy^{2}z^{3}\)")