Solve for a A = ninth root of a^4 square root of b* ninth root of b^4 square root of a

$A = \sqrt[9]{a^{4} \sqrt{b}} \cdot \sqrt[9]{b^{4} \sqrt{a}}$

Step 1

Rewrite the equation as $\sqrt[9]{a^{4} \sqrt{b}} \cdot \sqrt[9]{b^{4} \sqrt{a}} = A$ .

$\sqrt[9]{a^{4} \sqrt{b}} \cdot \sqrt[9]{b^{4} \sqrt{a}} = A$

Step 2

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $9$ .

${(\sqrt[9]{a^{4} \sqrt{b}} \cdot \sqrt[9]{b^{4} \sqrt{a}})}^{9} = A^{9}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{a^{4} \sqrt{b}}$ as ${(a^{4} \sqrt{b})}^{\frac{1}{9}}$ .

${({(a^{4} \sqrt{b})}^{\frac{1}{9}} \cdot \sqrt[9]{b^{4} \sqrt{a}})}^{9} = A^{9}$

Simplify the left side.

Simplify ${({(a^{4} \sqrt{b})}^{\frac{1}{9}} \cdot \sqrt[9]{b^{4} \sqrt{a}})}^{9}$ .

Apply the product rule to $a^{4} \sqrt{b}$ .

${({(a^{4})}^{\frac{1}{9}} {\sqrt{b}}^{\frac{1}{9}} \cdot \sqrt[9]{b^{4} \sqrt{a}})}^{9} = A^{9}$

Multiply the exponents in ${(a^{4})}^{\frac{1}{9}}$ .

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

${(a^{4 (\frac{1}{9})} {\sqrt{b}}^{\frac{1}{9}} \cdot \sqrt[9]{b^{4} \sqrt{a}})}^{9} = A^{9}$

Combine $4$ and $\frac{1}{9}$ .

${(a^{\frac{4}{9}} {\sqrt{b}}^{\frac{1}{9}} \cdot \sqrt[9]{b^{4} \sqrt{a}})}^{9} = A^{9}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $a^{\frac{4}{9}} {\sqrt{b}}^{\frac{1}{9}} \sqrt[9]{b^{4} \sqrt{a}}$ .

${(a^{\frac{4}{9}} {\sqrt{b}}^{\frac{1}{9}})}^{9} {\sqrt[9]{b^{4} \sqrt{a}}}^{9} = A^{9}$

Apply the product rule to $a^{\frac{4}{9}} {\sqrt{b}}^{\frac{1}{9}}$ .

${(a^{\frac{4}{9}})}^{9} {({\sqrt{b}}^{\frac{1}{9}})}^{9} {\sqrt[9]{b^{4} \sqrt{a}}}^{9} = A^{9}$

Multiply the exponents in ${(a^{\frac{4}{9}})}^{9}$ .

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

$a^{\frac{4}{9} \cdot 9} {({\sqrt{b}}^{\frac{1}{9}})}^{9} {\sqrt[9]{b^{4} \sqrt{a}}}^{9} = A^{9}$

Cancel the common factor of $9$ .

Cancel the common factor.

$a^{\frac{4}{9} \cdot 9} {({\sqrt{b}}^{\frac{1}{9}})}^{9} {\sqrt[9]{b^{4} \sqrt{a}}}^{9} = A^{9}$

Rewrite the expression.

$a^{4} {({\sqrt{b}}^{\frac{1}{9}})}^{9} {\sqrt[9]{b^{4} \sqrt{a}}}^{9} = A^{9}$

Multiply the exponents in ${({\sqrt{b}}^{\frac{1}{9}})}^{9}$ .

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

$a^{4} {\sqrt{b}}^{\frac{1}{9} \cdot 9} {\sqrt[9]{b^{4} \sqrt{a}}}^{9} = A^{9}$

Cancel the common factor of $9$ .

Cancel the common factor.

$a^{4} {\sqrt{b}}^{\frac{1}{9} \cdot 9} {\sqrt[9]{b^{4} \sqrt{a}}}^{9} = A^{9}$

Rewrite the expression.

$a^{4} {\sqrt{b}}^{1} {\sqrt[9]{b^{4} \sqrt{a}}}^{9} = A^{9}$

Simplify.

$a^{4} \sqrt{b} {\sqrt[9]{b^{4} \sqrt{a}}}^{9} = A^{9}$

Rewrite ${\sqrt[9]{b^{4} \sqrt{a}}}^{9}$ as $b^{4} \sqrt{a}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{b^{4} \sqrt{a}}$ as ${(b^{4} \sqrt{a})}^{\frac{1}{9}}$ .

$a^{4} \sqrt{b} {({(b^{4} \sqrt{a})}^{\frac{1}{9}})}^{9} = A^{9}$

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

$a^{4} \sqrt{b} {(b^{4} \sqrt{a})}^{\frac{1}{9} \cdot 9} = A^{9}$

Combine $\frac{1}{9}$ and $9$ .

$a^{4} \sqrt{b} {(b^{4} \sqrt{a})}^{\frac{9}{9}} = A^{9}$

Cancel the common factor of $9$ .

Cancel the common factor.

$a^{4} \sqrt{b} {(b^{4} \sqrt{a})}^{\frac{9}{9}} = A^{9}$

Rewrite the expression.

$a^{4} \sqrt{b} {(b^{4} \sqrt{a})}^{1} = A^{9}$

Simplify.

$a^{4} \sqrt{b} (b^{4} \sqrt{a}) = A^{9}$

Combine using the product rule for radicals.

$a^{4} (b^{4} \sqrt{b a}) = A^{9}$

Reorder factors in $a^{4} (b^{4} \sqrt{b a})$ .

$a^{4} b^{4} \sqrt{b a} = A^{9}$

Step 4

To remove the radical on the left side of the equation, square both sides of the equation.

${(a^{4} b^{4} \sqrt{b a})}^{2} = {(A^{9})}^{2}$

Step 5

Simplify each side of the equation.

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

${(a^{4} b^{4} {(b a)}^{\frac{1}{2}})}^{2} = {(A^{9})}^{2}$

Simplify the left side.

Simplify ${(a^{4} b^{4} {(b a)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $b a$ .

${(a^{4} b^{4} (b^{\frac{1}{2}} a^{\frac{1}{2}}))}^{2} = {(A^{9})}^{2}$

Multiply $a^{4}$ by $a^{\frac{1}{2}}$ by adding the exponents.

Move $a^{\frac{1}{2}}$ .

${(a^{\frac{1}{2}} a^{4} b^{4} b^{\frac{1}{2}})}^{2} = {(A^{9})}^{2}$

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

${(a^{\frac{1}{2} + 4} b^{4} b^{\frac{1}{2}})}^{2} = {(A^{9})}^{2}$

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(a^{\frac{1}{2} + 4 \cdot \frac{2}{2}} b^{4} b^{\frac{1}{2}})}^{2} = {(A^{9})}^{2}$

Combine $4$ and $\frac{2}{2}$ .

${(a^{\frac{1}{2} + \frac{4 \cdot 2}{2}} b^{4} b^{\frac{1}{2}})}^{2} = {(A^{9})}^{2}$

Combine the numerators over the common denominator.

${(a^{\frac{1 + 4 \cdot 2}{2}} b^{4} b^{\frac{1}{2}})}^{2} = {(A^{9})}^{2}$

Simplify the numerator.

Multiply $4$ by $2$ .

${(a^{\frac{1 + 8}{2}} b^{4} b^{\frac{1}{2}})}^{2} = {(A^{9})}^{2}$

Add $1$ and $8$ .

${(a^{\frac{9}{2}} b^{4} b^{\frac{1}{2}})}^{2} = {(A^{9})}^{2}$

Multiply $b^{4}$ by $b^{\frac{1}{2}}$ by adding the exponents.

Move $b^{\frac{1}{2}}$ .

${(a^{\frac{9}{2}} (b^{\frac{1}{2}} b^{4}))}^{2} = {(A^{9})}^{2}$

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

${(a^{\frac{9}{2}} b^{\frac{1}{2} + 4})}^{2} = {(A^{9})}^{2}$

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(a^{\frac{9}{2}} b^{\frac{1}{2} + 4 \cdot \frac{2}{2}})}^{2} = {(A^{9})}^{2}$

Combine $4$ and $\frac{2}{2}$ .

${(a^{\frac{9}{2}} b^{\frac{1}{2} + \frac{4 \cdot 2}{2}})}^{2} = {(A^{9})}^{2}$

Combine the numerators over the common denominator.

${(a^{\frac{9}{2}} b^{\frac{1 + 4 \cdot 2}{2}})}^{2} = {(A^{9})}^{2}$

Simplify the numerator.

Multiply $4$ by $2$ .

${(a^{\frac{9}{2}} b^{\frac{1 + 8}{2}})}^{2} = {(A^{9})}^{2}$

Add $1$ and $8$ .

${(a^{\frac{9}{2}} b^{\frac{9}{2}})}^{2} = {(A^{9})}^{2}$

Apply the product rule to $a^{\frac{9}{2}} b^{\frac{9}{2}}$ .

${(a^{\frac{9}{2}})}^{2} {(b^{\frac{9}{2}})}^{2} = {(A^{9})}^{2}$

Multiply the exponents in ${(a^{\frac{9}{2}})}^{2}$ .

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

$a^{\frac{9}{2} \cdot 2} {(b^{\frac{9}{2}})}^{2} = {(A^{9})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$a^{\frac{9}{2} \cdot 2} {(b^{\frac{9}{2}})}^{2} = {(A^{9})}^{2}$

Rewrite the expression.

$a^{9} {(b^{\frac{9}{2}})}^{2} = {(A^{9})}^{2}$

Multiply the exponents in ${(b^{\frac{9}{2}})}^{2}$ .

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

$a^{9} b^{\frac{9}{2} \cdot 2} = {(A^{9})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$a^{9} b^{\frac{9}{2} \cdot 2} = {(A^{9})}^{2}$

Rewrite the expression.

$a^{9} b^{9} = {(A^{9})}^{2}$

Simplify the right side.

Multiply the exponents in ${(A^{9})}^{2}$ .

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

$a^{9} b^{9} = A^{9 \cdot 2}$

Multiply $9$ by $2$ .

$a^{9} b^{9} = A^{18}$

Step 6

Solve for $a$ .

Divide each term in $a^{9} b^{9} = A^{18}$ by $b^{9}$ and simplify.

Divide each term in $a^{9} b^{9} = A^{18}$ by $b^{9}$ .

$\frac{a^{9} b^{9}}{b^{9}} = \frac{A^{18}}{b^{9}}$

Simplify the left side.

Cancel the common factor of $b^{9}$ .

Cancel the common factor.

$\frac{a^{9} b^{9}}{b^{9}} = \frac{A^{18}}{b^{9}}$

Divide $a^{9}$ by $1$ .

$a^{9} = \frac{A^{18}}{b^{9}}$

Take the 9th root of both sides of the equation to eliminate the exponent on the left side.

$a = \sqrt[9]{\frac{A^{18}}{b^{9}}}$

Simplify $\sqrt[9]{\frac{A^{18}}{b^{9}}}$ .

Rewrite $A^{18}$ as ${(A^{2})}^{9}$ .

$a = \sqrt[9]{\frac{{(A^{2})}^{9}}{b^{9}}}$

Rewrite $\frac{{(A^{2})}^{9}}{b^{9}}$ as ${(\frac{A^{2}}{b})}^{9}$ .

$a = \sqrt[9]{{(\frac{A^{2}}{b})}^{9}}$

Pull terms out from under the radical, assuming real numbers.

$a = \frac{A^{2}}{b}$

Solve for a A = ninth root of a^4 square root of b* ninth root of b^4 square root of a

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