Simplify fourth root of a^2b* cube root of ab

$\sqrt[4]{a^{2} b} \cdot \sqrt[3]{a b}$

Step 1

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

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

${(a^{2} b)}^{\frac{1}{4}} \sqrt[3]{a b}$

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

${(a^{2} b)}^{\frac{3}{12}} \sqrt[3]{a b}$

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

$\sqrt[12]{{(a^{2} b)}^{3}} \sqrt[3]{a b}$

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

$\sqrt[12]{{(a^{2} b)}^{3}} {(a b)}^{\frac{1}{3}}$

Rewrite ${(a b)}^{\frac{1}{3}}$ as ${(a b)}^{\frac{4}{12}}$ .

$\sqrt[12]{{(a^{2} b)}^{3}} {(a b)}^{\frac{4}{12}}$

Rewrite ${(a b)}^{\frac{4}{12}}$ as $\sqrt[12]{{(a b)}^{4}}$ .

$\sqrt[12]{{(a^{2} b)}^{3}} \sqrt[12]{{(a b)}^{4}}$

Step 2

Combine using the product rule for radicals.

$\sqrt[12]{{(a^{2} b)}^{3} {(a b)}^{4}}$

Step 3

Apply the product rule to $a^{2} b$ .

$\sqrt[12]{{(a^{2})}^{3} b^{3} {(a b)}^{4}}$

Step 4

Apply the product rule to $a b$ .

$\sqrt[12]{{(a^{2})}^{3} b^{3} a^{4} b^{4}}$

Step 5

Multiply $b^{3}$ by $b^{4}$ by adding the exponents.

Move $b^{4}$ .

$\sqrt[12]{{(a^{2})}^{3} (b^{4} b^{3}) a^{4}}$

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

$\sqrt[12]{{(a^{2})}^{3} b^{4 + 3} a^{4}}$

Add $4$ and $3$ .

$\sqrt[12]{{(a^{2})}^{3} b^{7} a^{4}}$

Step 6

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

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

$\sqrt[12]{a^{2 \cdot 3} b^{7} a^{4}}$

Multiply $2$ by $3$ .

$\sqrt[12]{a^{6} b^{7} a^{4}}$

Step 7

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

Move $a^{4}$ .

$\sqrt[12]{a^{4} a^{6} b^{7}}$

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

$\sqrt[12]{a^{4 + 6} b^{7}}$

Add $4$ and $6$ .

$\sqrt[12]{a^{10} b^{7}}$

Simplify fourth root of a^2b* cube root of ab

Related Questions

Support LCMGCF.com by disabling your adblocker.