Precalculus Examples

$\sqrt{9 u - 4} = \sqrt{7 u - 20}$

Step 1

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

${\sqrt{9 u - 4}}^{2} = {\sqrt{7 u - 20}}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 u - 4}$ as ${(9 u - 4)}^{\frac{1}{2}}$ .

${({(9 u - 4)}^{\frac{1}{2}})}^{2} = {\sqrt{7 u - 20}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(9 u - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(9 u - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(9 u - 4)}^{\frac{1}{2} \cdot 2} = {\sqrt{7 u - 20}}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(9 u - 4)}^{\frac{1}{2} \cdot 2} = {\sqrt{7 u - 20}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(9 u - 4)}^{1} = {\sqrt{7 u - 20}}^{2}$

Step 2.2.1.2

Simplify.

$9 u - 4 = {\sqrt{7 u - 20}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Rewrite ${\sqrt{7 u - 20}}^{2}$ as $7 u - 20$ .

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Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 u - 20}$ as ${(7 u - 20)}^{\frac{1}{2}}$ .

$9 u - 4 = {({(7 u - 20)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2

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

$9 u - 4 = {(7 u - 20)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3

Combine $\frac{1}{2}$ and $2$ .

$9 u - 4 = {(7 u - 20)}^{\frac{2}{2}}$

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Cancel the common factor.

$9 u - 4 = {(7 u - 20)}^{\frac{2}{2}}$

Step 2.3.1.4.2

Rewrite the expression.

$9 u - 4 = {(7 u - 20)}^{1}$

Step 2.3.1.5

Simplify.

$9 u - 4 = 7 u - 20$

Step 3

Solve for $u$ .

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Step 3.1

Move all terms containing $u$ to the left side of the equation.

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Step 3.1.1

Subtract $7 u$ from both sides of the equation.

$9 u - 4 - 7 u = - 20$

Step 3.1.2

Subtract $7 u$ from $9 u$ .

$2 u - 4 = - 20$

Step 3.2

Move all terms not containing $u$ to the right side of the equation.

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Step 3.2.1

Add $4$ to both sides of the equation.

$2 u = - 20 + 4$

Step 3.2.2

Add $- 20$ and $4$ .

$2 u = - 16$

Step 3.3

Divide each term in $2 u = - 16$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 u = - 16$ by $2$ .

$\frac{2 u}{2} = \frac{- 16}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{- 16}{2}$

Step 3.3.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 16}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 16$ by $2$ .

$u = - 8$

Step 4

Exclude the solutions that do not make $\sqrt{9 u - 4} = \sqrt{7 u - 20}$ true.

$No solution$