Question

try it!
try: determine equivalent expressions
determine whether each of the radical expressions below is equivalent to
$x^{\frac{3}{2}} y^{\frac{1}{3}}$.

equivalent not equivalent

$\sqrt{x^3} \cdot \sqrt3{y}$ $\bullet$ $\circ$
$\sqrt{xy}$ $\circ$ $\circ$
$\sqrt{x^3 y}$ $\circ$ $\circ$
$\sqrt6{x^9 y^2}$ $\circ$ $\circ$

check
explain

Explanation:

Step1: Recall radical-exponent rules

Recall that \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\) and \(a^m\cdot a^n = a^{m + n}\), also \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\).

Step2: Analyze \(\sqrt{x^3}\cdot\sqrt[3]{y}\)

For \(\sqrt{x^3}\), using \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), here \(n = 2\), \(m=3\), so \(\sqrt{x^3}=x^{\frac{3}{2}}\). For \(\sqrt[3]{y}\), \(n = 3\), \(m = 1\), so \(\sqrt[3]{y}=y^{\frac{1}{3}}\). Then \(\sqrt{x^3}\cdot\sqrt[3]{y}=x^{\frac{3}{2}}y^{\frac{1}{3}}\), which matches the given expression \(x^{\frac{3}{2}}y^{\frac{1}{3}}\), so it is equivalent.

Step3: Analyze \(\sqrt{xy}\)

\(\sqrt{xy}=(xy)^{\frac{1}{2}}=x^{\frac{1}{2}}y^{\frac{1}{2}}\), which is not equal to \(x^{\frac{3}{2}}y^{\frac{1}{3}}\) (exponents of \(x\) and \(y\) don't match), so not equivalent.

Step4: Analyze \(\sqrt{x^3y}\)

\(\sqrt{x^3y}=(x^3y)^{\frac{1}{2}}=x^{\frac{3}{2}}y^{\frac{1}{2}}\), exponent of \(y\) is \(\frac{1}{2}\) vs \(\frac{1}{3}\) in given, so not equivalent.

Step5: Analyze \(\sqrt[6]{x^9y^2}\)

Using \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), \(n = 6\), \(m = 9\) for \(x\): \(x^{\frac{9}{6}}=x^{\frac{3}{2}}\); for \(y\), \(m = 2\), \(n = 6\): \(y^{\frac{2}{6}}=y^{\frac{1}{3}}\). So \(\sqrt[6]{x^9y^2}=x^{\frac{3}{2}}y^{\frac{1}{3}}\), which matches the given expression, so it is equivalent.

Answer:

• \(\boldsymbol{\sqrt{x^3}\cdot\sqrt[3]{y}}\): Equivalent
• \(\boldsymbol{\sqrt{xy}}\): Not equivalent
• \(\boldsymbol{\sqrt{x^3y}}\): Not equivalent
• \(\boldsymbol{\sqrt[6]{x^9y^2}}\): Equivalent

(For the checkboxes:

• \(\sqrt{x^3}\cdot\sqrt[3]{y}\): Select "Equivalent" (already selected as black dot)
• \(\sqrt{xy}\): Select "Not equivalent"
• \(\sqrt{x^3y}\): Select "Not equivalent"
• \(\sqrt[6]{x^9y^2}\): Select "Equivalent")