Basic Math Examples

$\sqrt[5]{r^{2} t} \cdot \sqrt[3]{r t}$

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 $\sqrt[5]{r^{2} t}$ as ${(r^{2} t)}^{\frac{1}{5}}$ .

${(r^{2} t)}^{\frac{1}{5}} \sqrt[3]{r t}$

Step 1.2

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

${(r^{2} t)}^{\frac{3}{15}} \sqrt[3]{r t}$

Step 1.3

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

$\sqrt[15]{{(r^{2} t)}^{3}} \sqrt[3]{r t}$

Step 1.4

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

$\sqrt[15]{{(r^{2} t)}^{3}} {(r t)}^{\frac{1}{3}}$

Step 1.5

Rewrite ${(r t)}^{\frac{1}{3}}$ as ${(r t)}^{\frac{5}{15}}$ .

$\sqrt[15]{{(r^{2} t)}^{3}} {(r t)}^{\frac{5}{15}}$

Step 1.6

Rewrite ${(r t)}^{\frac{5}{15}}$ as $\sqrt[15]{{(r t)}^{5}}$ .

$\sqrt[15]{{(r^{2} t)}^{3}} \sqrt[15]{{(r t)}^{5}}$

Step 2

Combine using the product rule for radicals.

$\sqrt[15]{{(r^{2} t)}^{3} {(r t)}^{5}}$

Step 3

Apply the product rule to $r^{2} t$ .

$\sqrt[15]{{(r^{2})}^{3} t^{3} {(r t)}^{5}}$

Step 4

Apply the product rule to $r t$ .

$\sqrt[15]{{(r^{2})}^{3} t^{3} r^{5} t^{5}}$

Step 5

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

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Step 5.1

Move $t^{5}$ .

$\sqrt[15]{{(r^{2})}^{3} (t^{5} t^{3}) r^{5}}$

Step 5.2

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

$\sqrt[15]{{(r^{2})}^{3} t^{5 + 3} r^{5}}$

Step 5.3

Add $5$ and $3$ .

$\sqrt[15]{{(r^{2})}^{3} t^{8} r^{5}}$

Step 6

Multiply the exponents in ${(r^{2})}^{3}$ .

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Step 6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt[15]{r^{2 \cdot 3} t^{8} r^{5}}$

Step 6.2

Multiply $2$ by $3$ .

$\sqrt[15]{r^{6} t^{8} r^{5}}$

Step 7

Multiply $r^{6}$ by $r^{5}$ by adding the exponents.

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Step 7.1

Move $r^{5}$ .

$\sqrt[15]{r^{5} r^{6} t^{8}}$

Step 7.2

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

$\sqrt[15]{r^{5 + 6} t^{8}}$

Step 7.3

Add $5$ and $6$ .

$\sqrt[15]{r^{11} t^{8}}$