Algebra Examples

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

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.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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Step 3.2.1.2.1

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

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

Step 3.2.1.2.2

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

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

Step 3.2.1.3

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

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Step 3.2.1.3.1

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

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

Step 3.2.1.3.2

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

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

Step 3.2.1.4

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

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Step 3.2.1.4.1

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

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

Step 3.2.1.4.2

Cancel the common factor of $9$ .

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Step 3.2.1.4.2.1

Cancel the common factor.

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

Step 3.2.1.4.2.2

Rewrite the expression.

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

Step 3.2.1.5

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

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Step 3.2.1.5.1

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

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

Step 3.2.1.5.2

Cancel the common factor of $9$ .

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Step 3.2.1.5.2.1

Cancel the common factor.

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

Step 3.2.1.5.2.2

Rewrite the expression.

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

Step 3.2.1.6

Simplify.

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

Step 3.2.1.7

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

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Step 3.2.1.7.1

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

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

Step 3.2.1.7.2

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

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

Step 3.2.1.7.3

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

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

Step 3.2.1.7.4

Cancel the common factor of $9$ .

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Step 3.2.1.7.4.1

Cancel the common factor.

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

Step 3.2.1.7.4.2

Rewrite the expression.

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

Step 3.2.1.7.5

Simplify.

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

Step 3.2.1.8

Combine using the product rule for radicals.

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

Step 3.2.1.9

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.

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

Apply the product rule to $b a$ .

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

Step 5.2.1.2

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

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Step 5.2.1.2.1

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

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

Step 5.2.1.2.2

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

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

Step 5.2.1.2.3

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

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

Step 5.2.1.2.4

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

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

Step 5.2.1.2.5

Combine the numerators over the common denominator.

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

Step 5.2.1.2.6

Simplify the numerator.

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Step 5.2.1.2.6.1

Multiply $4$ by $2$ .

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

Step 5.2.1.2.6.2

Add $1$ and $8$ .

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

Step 5.2.1.3

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

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Step 5.2.1.3.1

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

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

Step 5.2.1.3.2

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

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

Step 5.2.1.3.3

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

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

Step 5.2.1.3.4

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

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

Step 5.2.1.3.5

Combine the numerators over the common denominator.

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

Step 5.2.1.3.6

Simplify the numerator.

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Step 5.2.1.3.6.1

Multiply $4$ by $2$ .

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

Step 5.2.1.3.6.2

Add $1$ and $8$ .

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

Step 5.2.1.4

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

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

Step 5.2.1.5

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

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Step 5.2.1.5.1

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

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

Step 5.2.1.5.2

Cancel the common factor of $2$ .

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Step 5.2.1.5.2.1

Cancel the common factor.

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

Step 5.2.1.5.2.2

Rewrite the expression.

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

Step 5.2.1.6

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

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Step 5.2.1.6.1

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

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

Step 5.2.1.6.2

Cancel the common factor of $2$ .

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Step 5.2.1.6.2.1

Cancel the common factor.

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

Step 5.2.1.6.2.2

Rewrite the expression.

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

Step 5.3

Simplify the right side.

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Step 5.3.1

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

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Step 5.3.1.1

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

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

Step 5.3.1.2

Multiply $9$ by $2$ .

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

Step 6

Solve for $a$ .

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

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

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Step 6.1.2.1.1

Cancel the common factor.

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

Step 6.1.2.1.2

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

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

Step 6.2

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

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

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

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

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

Step 6.3.3

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

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