Algebra Examples

$\sqrt{\frac{x}{7}} = \sqrt{24 - x}$

Step 1

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

${\sqrt{\frac{x}{7}}}^{2} = {\sqrt{24 - x}}^{2}$

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

${({(\frac{x}{7})}^{\frac{1}{2}})}^{2} = {\sqrt{24 - x}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(\frac{x}{7})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(\frac{x}{7})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(\frac{x}{7})}^{\frac{1}{2} \cdot 2} = {\sqrt{24 - x}}^{2}$

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.

${(\frac{x}{7})}^{\frac{1}{2} \cdot 2} = {\sqrt{24 - x}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(\frac{x}{7})}^{1} = {\sqrt{24 - x}}^{2}$

Step 2.2.1.2

Simplify.

$\frac{x}{7} = {\sqrt{24 - x}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Rewrite ${\sqrt{24 - x}}^{2}$ as $24 - x$ .

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Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{24 - x}$ as ${(24 - x)}^{\frac{1}{2}}$ .

$\frac{x}{7} = {({(24 - x)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2

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

$\frac{x}{7} = {(24 - x)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3

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

$\frac{x}{7} = {(24 - x)}^{\frac{2}{2}}$

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.

$\frac{x}{7} = {(24 - x)}^{\frac{2}{2}}$

Step 2.3.1.4.2

Rewrite the expression.

$\frac{x}{7} = {(24 - x)}^{1}$

Step 2.3.1.5

Simplify.

$\frac{x}{7} = 24 - x$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $7$ .

$\frac{x}{7} \cdot 7 = (24 - x) \cdot 7$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $7$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{x}{7} \cdot 7 = (24 - x) \cdot 7$

Step 3.2.1.1.2

Rewrite the expression.

$x = (24 - x) \cdot 7$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(24 - x) \cdot 7$ .

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Step 3.2.2.1.1

Apply the distributive property.

$x = 24 \cdot 7 - x \cdot 7$

Step 3.2.2.1.2

Simplify the expression.

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Step 3.2.2.1.2.1

Multiply $24$ by $7$ .

$x = 168 - x \cdot 7$

Step 3.2.2.1.2.2

Multiply $7$ by $- 1$ .

$x = 168 - 7 x$

Step 3.2.2.1.2.3

Reorder $168$ and $- 7 x$ .

$x = - 7 x + 168$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

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Step 3.3.1.1

Add $7 x$ to both sides of the equation.

$x + 7 x = 168$

Step 3.3.1.2

Add $x$ and $7 x$ .

$8 x = 168$

Step 3.3.2

Divide each term in $8 x = 168$ by $8$ and simplify.

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Step 3.3.2.1

Divide each term in $8 x = 168$ by $8$ .

$\frac{8 x}{8} = \frac{168}{8}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $8$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{168}{8}$

Step 3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{168}{8}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $168$ by $8$ .

$x = 21$