Pre-Algebra Examples

$\sqrt[5]{2 m n} \sqrt[3]{3 m n}$

Step 1

Rewrite the expression using the least common index of $15$ .

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{2 m n}$ as ${(2 m n)}^{\frac{1}{5}}$ .

${(2 m n)}^{\frac{1}{5}} \sqrt[3]{3 m n}$

Step 1.2

Rewrite ${(2 m n)}^{\frac{1}{5}}$ as ${(2 m n)}^{\frac{3}{15}}$ .

${(2 m n)}^{\frac{3}{15}} \sqrt[3]{3 m n}$

Step 1.3

Rewrite ${(2 m n)}^{\frac{3}{15}}$ as $\sqrt[15]{{(2 m n)}^{3}}$ .

$\sqrt[15]{{(2 m n)}^{3}} \sqrt[3]{3 m n}$

Step 1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{3 m n}$ as ${(3 m n)}^{\frac{1}{3}}$ .

$\sqrt[15]{{(2 m n)}^{3}} {(3 m n)}^{\frac{1}{3}}$

Step 1.5

Rewrite ${(3 m n)}^{\frac{1}{3}}$ as ${(3 m n)}^{\frac{5}{15}}$ .

$\sqrt[15]{{(2 m n)}^{3}} {(3 m n)}^{\frac{5}{15}}$

Step 1.6

Rewrite ${(3 m n)}^{\frac{5}{15}}$ as $\sqrt[15]{{(3 m n)}^{5}}$ .

$\sqrt[15]{{(2 m n)}^{3}} \sqrt[15]{{(3 m n)}^{5}}$

Step 2

Combine using the product rule for radicals.

$\sqrt[15]{{(2 m n)}^{3} {(3 m n)}^{5}}$

Step 3

Apply the product rule to $2 m n$ .

$\sqrt[15]{{(2 m)}^{3} n^{3} {(3 m n)}^{5}}$

Step 4

Apply the product rule to $3 m n$ .

$\sqrt[15]{{(2 m)}^{3} n^{3} {(3 m)}^{5} n^{5}}$

Step 5

Multiply $n^{3}$ by $n^{5}$ by adding the exponents.

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Step 5.1

Move $n^{5}$ .

$\sqrt[15]{{(2 m)}^{3} (n^{5} n^{3}) {(3 m)}^{5}}$

Step 5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt[15]{{(2 m)}^{3} n^{5 + 3} {(3 m)}^{5}}$

Step 5.3

Add $5$ and $3$ .

$\sqrt[15]{{(2 m)}^{3} n^{8} {(3 m)}^{5}}$

Step 6

Apply the product rule to $2 m$ .

$\sqrt[15]{2^{3} m^{3} n^{8} {(3 m)}^{5}}$

Step 7

Raise $2$ to the power of $3$ .

$\sqrt[15]{8 m^{3} n^{8} {(3 m)}^{5}}$

Step 8

Apply the product rule to $3 m$ .

$\sqrt[15]{8 m^{3} n^{8} (3^{5} m^{5})}$

Step 9

Raise $3$ to the power of $5$ .

$\sqrt[15]{8 m^{3} n^{8} \cdot 243 m^{5}}$

Step 10

Combine exponents.

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Step 10.1

Multiply $243$ by $8$ .

$\sqrt[15]{1944 m^{3} n^{8} m^{5}}$

Step 10.2

Multiply $m^{3}$ by $m^{5}$ by adding the exponents.

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Step 10.2.1

Move $m^{5}$ .

$\sqrt[15]{1944 (m^{5} m^{3}) n^{8}}$

Step 10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt[15]{1944 m^{5 + 3} n^{8}}$

Step 10.2.3

Add $5$ and $3$ .

$\sqrt[15]{1944 m^{8} n^{8}}$