Evaluate ( square root of a square root of b)/( seventh root of ab)

$\frac{\sqrt{a \sqrt{b}}}{\sqrt[7]{a b}}$

Step 1

Multiply $\frac{\sqrt{a \sqrt{b}}}{\sqrt[7]{a b}}$ by $\frac{{\sqrt[7]{a b}}^{6}}{{\sqrt[7]{a b}}^{6}}$ .

$\frac{\sqrt{a \sqrt{b}}}{\sqrt[7]{a b}} \cdot \frac{{\sqrt[7]{a b}}^{6}}{{\sqrt[7]{a b}}^{6}}$

Step 2

Combine and simplify the denominator.

Multiply $\frac{\sqrt{a \sqrt{b}}}{\sqrt[7]{a b}}$ by $\frac{{\sqrt[7]{a b}}^{6}}{{\sqrt[7]{a b}}^{6}}$ .

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

Raise $\sqrt[7]{a b}$ to the power of $1$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[7]{a b}}^{6}}{{\sqrt[7]{a b}}^{1} {\sqrt[7]{a b}}^{6}}$

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

$\frac{\sqrt{a \sqrt{b}} {\sqrt[7]{a b}}^{6}}{{\sqrt[7]{a b}}^{1 + 6}}$

Add $1$ and $6$ .

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

Rewrite ${\sqrt[7]{a b}}^{7}$ as $a b$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{a b}$ as ${(a b)}^{\frac{1}{7}}$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[7]{a b}}^{6}}{{({(a b)}^{\frac{1}{7}})}^{7}}$

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

$\frac{\sqrt{a \sqrt{b}} {\sqrt[7]{a b}}^{6}}{{(a b)}^{\frac{1}{7} \cdot 7}}$

Combine $\frac{1}{7}$ and $7$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[7]{a b}}^{6}}{{(a b)}^{\frac{7}{7}}}$

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{\sqrt{a \sqrt{b}} {\sqrt[7]{a b}}^{6}}{{(a b)}^{\frac{7}{7}}}$

Rewrite the expression.

$\frac{\sqrt{a \sqrt{b}} {\sqrt[7]{a b}}^{6}}{{(a b)}^{1}}$

Simplify.

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

Step 3

Simplify the numerator.

Rewrite ${\sqrt[7]{a b}}^{6}$ as ${({(a b)}^{6})}^{\frac{1}{7}}$ .

$\frac{\sqrt{a \sqrt{b}} \sqrt[7]{{(a b)}^{6}}}{a b}$

Apply the product rule to $a b$ .

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

Step 4

Simplify the numerator.

Rewrite the expression using the least common index of $14$ .

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

$\frac{{(a \sqrt{b})}^{\frac{1}{2}} \sqrt[7]{a^{6} b^{6}}}{a b}$

Rewrite ${(a \sqrt{b})}^{\frac{1}{2}}$ as ${(a \sqrt{b})}^{\frac{7}{14}}$ .

$\frac{{(a \sqrt{b})}^{\frac{7}{14}} \sqrt[7]{a^{6} b^{6}}}{a b}$

Rewrite ${(a \sqrt{b})}^{\frac{7}{14}}$ as $\sqrt[14]{{(a \sqrt{b})}^{7}}$ .

$\frac{\sqrt[14]{{(a \sqrt{b})}^{7}} \sqrt[7]{a^{6} b^{6}}}{a b}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{a^{6} b^{6}}$ as ${(a^{6} b^{6})}^{\frac{1}{7}}$ .

$\frac{\sqrt[14]{{(a \sqrt{b})}^{7}} {(a^{6} b^{6})}^{\frac{1}{7}}}{a b}$

Rewrite ${(a^{6} b^{6})}^{\frac{1}{7}}$ as ${(a^{6} b^{6})}^{\frac{2}{14}}$ .

$\frac{\sqrt[14]{{(a \sqrt{b})}^{7}} {(a^{6} b^{6})}^{\frac{2}{14}}}{a b}$

Rewrite ${(a^{6} b^{6})}^{\frac{2}{14}}$ as $\sqrt[14]{{(a^{6} b^{6})}^{2}}$ .

$\frac{\sqrt[14]{{(a \sqrt{b})}^{7}} \sqrt[14]{{(a^{6} b^{6})}^{2}}}{a b}$

Combine using the product rule for radicals.

$\frac{\sqrt[14]{{(a \sqrt{b})}^{7} {(a^{6} b^{6})}^{2}}}{a b}$

Apply the product rule to $a \sqrt{b}$ .

$\frac{\sqrt[14]{a^{7} {\sqrt{b}}^{7} {(a^{6} b^{6})}^{2}}}{a b}$

Apply the product rule to $a^{6} b^{6}$ .

$\frac{\sqrt[14]{a^{7} {\sqrt{b}}^{7} {(a^{6})}^{2} {(b^{6})}^{2}}}{a b}$

Rewrite ${\sqrt{b}}^{7}$ as ${(b^{7})}^{\frac{1}{2}}$ .

$\frac{\sqrt[14]{a^{7} \sqrt{b^{7}} {(a^{6})}^{2} {(b^{6})}^{2}}}{a b}$

Rewrite $b^{7}$ as ${(b^{3})}^{2} b$ .

Factor out $b^{6}$ .

$\frac{\sqrt[14]{a^{7} \sqrt{b^{6} b} {(a^{6})}^{2} {(b^{6})}^{2}}}{a b}$

Rewrite $b^{6}$ as ${(b^{3})}^{2}$ .

$\frac{\sqrt[14]{a^{7} \sqrt{{(b^{3})}^{2} b} {(a^{6})}^{2} {(b^{6})}^{2}}}{a b}$

Pull terms out from under the radical.

$\frac{\sqrt[14]{a^{7} (b^{3} \sqrt{b}) {(a^{6})}^{2} {(b^{6})}^{2}}}{a b}$

Multiply the exponents in ${(a^{6})}^{2}$ .

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

$\frac{\sqrt[14]{a^{7} (b^{3} \sqrt{b}) a^{6 \cdot 2} {(b^{6})}^{2}}}{a b}$

Multiply $6$ by $2$ .

$\frac{\sqrt[14]{a^{7} (b^{3} \sqrt{b}) a^{12} {(b^{6})}^{2}}}{a b}$

Multiply the exponents in ${(b^{6})}^{2}$ .

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

$\frac{\sqrt[14]{a^{7} (b^{3} \sqrt{b}) a^{12} b^{6 \cdot 2}}}{a b}$

Multiply $6$ by $2$ .

$\frac{\sqrt[14]{a^{7} (b^{3} \sqrt{b}) a^{12} b^{12}}}{a b}$

$\frac{\sqrt[14]{a^{7} b^{3} \sqrt{b} a^{12} b^{12}}}{a b}$

Combine exponents.

Multiply $a^{7}$ by $a^{12}$ by adding the exponents.

Move $a^{12}$ .

$\frac{\sqrt[14]{a^{12} a^{7} b^{3} \sqrt{b} b^{12}}}{a b}$

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

$\frac{\sqrt[14]{a^{12 + 7} b^{3} \sqrt{b} b^{12}}}{a b}$

Add $12$ and $7$ .

$\frac{\sqrt[14]{a^{19} b^{3} \sqrt{b} b^{12}}}{a b}$

Multiply $b^{3}$ by $b^{12}$ by adding the exponents.

Move $b^{12}$ .

$\frac{\sqrt[14]{a^{19} (b^{12} b^{3}) \sqrt{b}}}{a b}$

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

$\frac{\sqrt[14]{a^{19} b^{12 + 3} \sqrt{b}}}{a b}$

Add $12$ and $3$ .

$\frac{\sqrt[14]{a^{19} b^{15} \sqrt{b}}}{a b}$

Step 5

Simplify the numerator.

Rewrite $a^{19} b^{15} \sqrt{b}$ as ${(a b)}^{14} (a^{5} b \sqrt{b})$ .

Factor out $a^{14}$ .

$\frac{\sqrt[14]{a^{14} a^{5} b^{15} \sqrt{b}}}{a b}$

Factor out $b^{14}$ .

$\frac{\sqrt[14]{a^{14} a^{5} (b^{14} b) \sqrt{b}}}{a b}$

Move $a^{5}$ .

$\frac{\sqrt[14]{a^{14} b^{14} a^{5} b \sqrt{b}}}{a b}$

Rewrite $a^{14} b^{14}$ as ${(a b)}^{14}$ .

$\frac{\sqrt[14]{{(a b)}^{14} a^{5} b \sqrt{b}}}{a b}$

Add parentheses.

$\frac{\sqrt[14]{{(a b)}^{14} a^{5} (b \sqrt{b})}}{a b}$

Add parentheses.

$\frac{\sqrt[14]{{(a b)}^{14} (a^{5} b \sqrt{b})}}{a b}$

Pull terms out from under the radical.

$\frac{a b \sqrt[14]{a^{5} b \sqrt{b}}}{a b}$

Step 6

Reduce the expression by cancelling the common factors.

Cancel the common factor of $a$ .

Cancel the common factor.

$\frac{a b \sqrt[14]{a^{5} b \sqrt{b}}}{a b}$

Rewrite the expression.

$\frac{b \sqrt[14]{a^{5} b \sqrt{b}}}{b}$

Cancel the common factor of $b$ .

Cancel the common factor.

$\frac{b \sqrt[14]{a^{5} b \sqrt{b}}}{b}$

Divide $\sqrt[14]{a^{5} b \sqrt{b}}$ by $1$ .

$\sqrt[14]{a^{5} b \sqrt{b}}$

Evaluate ( square root of a square root of b)/( seventh root of ab)

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