Finite Math Examples

$\sqrt[3]{3 x y^{2}} \cdot \sqrt[4]{3 x^{2} y}$

Step 1

Rewrite the expression using the least common index of $12$ .

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{3 x y^{2}}$ as ${(3 x y^{2})}^{\frac{1}{3}}$ .

${(3 x y^{2})}^{\frac{1}{3}} \sqrt[4]{3 x^{2} y}$

Step 1.2

Rewrite ${(3 x y^{2})}^{\frac{1}{3}}$ as ${(3 x y^{2})}^{\frac{4}{12}}$ .

${(3 x y^{2})}^{\frac{4}{12}} \sqrt[4]{3 x^{2} y}$

Step 1.3

Rewrite ${(3 x y^{2})}^{\frac{4}{12}}$ as $\sqrt[12]{{(3 x y^{2})}^{4}}$ .

$\sqrt[12]{{(3 x y^{2})}^{4}} \sqrt[4]{3 x^{2} y}$

Step 1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3 x^{2} y}$ as ${(3 x^{2} y)}^{\frac{1}{4}}$ .

$\sqrt[12]{{(3 x y^{2})}^{4}} {(3 x^{2} y)}^{\frac{1}{4}}$

Step 1.5

Rewrite ${(3 x^{2} y)}^{\frac{1}{4}}$ as ${(3 x^{2} y)}^{\frac{3}{12}}$ .

$\sqrt[12]{{(3 x y^{2})}^{4}} {(3 x^{2} y)}^{\frac{3}{12}}$

Step 1.6

Rewrite ${(3 x^{2} y)}^{\frac{3}{12}}$ as $\sqrt[12]{{(3 x^{2} y)}^{3}}$ .

$\sqrt[12]{{(3 x y^{2})}^{4}} \sqrt[12]{{(3 x^{2} y)}^{3}}$

Step 2

Combine using the product rule for radicals.

$\sqrt[12]{{(3 x y^{2})}^{4} {(3 x^{2} y)}^{3}}$

Step 3

Apply the product rule to $3 x y^{2}$ .

$\sqrt[12]{{(3 x)}^{4} {(y^{2})}^{4} {(3 x^{2} y)}^{3}}$

Step 4

Apply the product rule to $3 x^{2} y$ .

$\sqrt[12]{{(3 x)}^{4} {(y^{2})}^{4} {(3 x^{2})}^{3} y^{3}}$

Step 5

Apply the product rule to $3 x$ .

$\sqrt[12]{3^{4} x^{4} {(y^{2})}^{4} {(3 x^{2})}^{3} y^{3}}$

Step 6

Raise $3$ to the power of $4$ .

$\sqrt[12]{81 x^{4} {(y^{2})}^{4} {(3 x^{2})}^{3} y^{3}}$

Step 7

Multiply the exponents in ${(y^{2})}^{4}$ .

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Step 7.1

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

$\sqrt[12]{81 x^{4} y^{2 \cdot 4} {(3 x^{2})}^{3} y^{3}}$

Step 7.2

Multiply $2$ by $4$ .

$\sqrt[12]{81 x^{4} y^{8} {(3 x^{2})}^{3} y^{3}}$

Step 8

Apply the product rule to $3 x^{2}$ .

$\sqrt[12]{81 x^{4} y^{8} (3^{3} {(x^{2})}^{3}) y^{3}}$

Step 9

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

$\sqrt[12]{81 x^{4} y^{8} (27 {(x^{2})}^{3}) y^{3}}$

Step 10

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

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Step 10.1

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

$\sqrt[12]{81 x^{4} y^{8} (27 x^{2 \cdot 3}) y^{3}}$

Step 10.2

Multiply $2$ by $3$ .

$\sqrt[12]{81 x^{4} y^{8} (27 x^{6}) y^{3}}$

$\sqrt[12]{81 x^{4} y^{8} \cdot 27 x^{6} y^{3}}$

Step 11

Combine exponents.

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Step 11.1

Multiply $27$ by $81$ .

$\sqrt[12]{2187 x^{4} y^{8} x^{6} y^{3}}$

Step 11.2

Multiply $x^{4}$ by $x^{6}$ by adding the exponents.

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Step 11.2.1

Move $x^{6}$ .

$\sqrt[12]{2187 (x^{6} x^{4}) y^{8} y^{3}}$

Step 11.2.2

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

$\sqrt[12]{2187 x^{6 + 4} y^{8} y^{3}}$

Step 11.2.3

Add $6$ and $4$ .

$\sqrt[12]{2187 x^{10} y^{8} y^{3}}$

Step 11.3

Multiply $y^{8}$ by $y^{3}$ by adding the exponents.

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Step 11.3.1

Move $y^{3}$ .

$\sqrt[12]{2187 x^{10} (y^{3} y^{8})}$

Step 11.3.2

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

$\sqrt[12]{2187 x^{10} y^{3 + 8}}$

Step 11.3.3

Add $3$ and $8$ .

$\sqrt[12]{2187 x^{10} y^{11}}$