Algebra Examples

$\sqrt{72 - x} = \sqrt{\frac{x}{5}}$

Step 1

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

${\sqrt{72 - x}}^{2} = {\sqrt{\frac{x}{5}}}^{2}$

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 $\sqrt{72 - x}$ as ${(72 - x)}^{\frac{1}{2}}$ .

${({(72 - x)}^{\frac{1}{2}})}^{2} = {\sqrt{\frac{x}{5}}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(72 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(72 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(72 - x)}^{\frac{1}{2} \cdot 2} = {\sqrt{\frac{x}{5}}}^{2}$

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.

${(72 - x)}^{\frac{1}{2} \cdot 2} = {\sqrt{\frac{x}{5}}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(72 - x)}^{1} = {\sqrt{\frac{x}{5}}}^{2}$

Step 2.2.1.2

Simplify.

$72 - x = {\sqrt{\frac{x}{5}}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${\sqrt{\frac{x}{5}}}^{2}$ .

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Step 2.3.1.1

Rewrite $\sqrt{\frac{x}{5}}$ as $\frac{\sqrt{x}}{\sqrt{5}}$ .

$72 - x = {(\frac{\sqrt{x}}{\sqrt{5}})}^{2}$

Step 2.3.1.2

Multiply $\frac{\sqrt{x}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$72 - x = {(\frac{\sqrt{x}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})}^{2}$

Step 2.3.1.3

Simplify terms.

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Step 2.3.1.3.1

Combine and simplify the denominator.

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Step 2.3.1.3.1.1

Multiply $\frac{\sqrt{x}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$72 - x = {(\frac{\sqrt{x} \sqrt{5}}{\sqrt{5} \sqrt{5}})}^{2}$

Step 2.3.1.3.1.2

Raise $\sqrt{5}$ to the power of $1$ .

$72 - x = {(\frac{\sqrt{x} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}})}^{2}$

Step 2.3.1.3.1.3

Raise $\sqrt{5}$ to the power of $1$ .

$72 - x = {(\frac{\sqrt{x} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}})}^{2}$

Step 2.3.1.3.1.4

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

$72 - x = {(\frac{\sqrt{x} \sqrt{5}}{{\sqrt{5}}^{1 + 1}})}^{2}$

Step 2.3.1.3.1.5

Add $1$ and $1$ .

$72 - x = {(\frac{\sqrt{x} \sqrt{5}}{{\sqrt{5}}^{2}})}^{2}$

Step 2.3.1.3.1.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.3.1.3.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$72 - x = {(\frac{\sqrt{x} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}})}^{2}$

Step 2.3.1.3.1.6.2

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

$72 - x = {(\frac{\sqrt{x} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}})}^{2}$

Step 2.3.1.3.1.6.3

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

$72 - x = {(\frac{\sqrt{x} \sqrt{5}}{5^{\frac{2}{2}}})}^{2}$

Step 2.3.1.3.1.6.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.6.4.1

Cancel the common factor.

$72 - x = {(\frac{\sqrt{x} \sqrt{5}}{5^{\frac{2}{2}}})}^{2}$

Step 2.3.1.3.1.6.4.2

Rewrite the expression.

$72 - x = {(\frac{\sqrt{x} \sqrt{5}}{5^{1}})}^{2}$

Step 2.3.1.3.1.6.5

Evaluate the exponent.

$72 - x = {(\frac{\sqrt{x} \sqrt{5}}{5})}^{2}$

Step 2.3.1.3.2

Combine using the product rule for radicals.

$72 - x = {(\frac{\sqrt{x \cdot 5}}{5})}^{2}$

Step 2.3.1.3.3

Apply the product rule to $\frac{\sqrt{x \cdot 5}}{5}$ .

$72 - x = \frac{{\sqrt{x \cdot 5}}^{2}}{5^{2}}$

Step 2.3.1.4

Simplify the numerator.

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Step 2.3.1.4.1

Rewrite ${\sqrt{x \cdot 5}}^{2}$ as $x \cdot 5$ .

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Step 2.3.1.4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x \cdot 5}$ as ${(x \cdot 5)}^{\frac{1}{2}}$ .

$72 - x = \frac{{({(x \cdot 5)}^{\frac{1}{2}})}^{2}}{5^{2}}$

Step 2.3.1.4.1.2

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

$72 - x = \frac{{(x \cdot 5)}^{\frac{1}{2} \cdot 2}}{5^{2}}$

Step 2.3.1.4.1.3

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

$72 - x = \frac{{(x \cdot 5)}^{\frac{2}{2}}}{5^{2}}$

Step 2.3.1.4.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1.4.1

Cancel the common factor.

$72 - x = \frac{{(x \cdot 5)}^{\frac{2}{2}}}{5^{2}}$

Step 2.3.1.4.1.4.2

Rewrite the expression.

$72 - x = \frac{{(x \cdot 5)}^{1}}{5^{2}}$

Step 2.3.1.4.1.5

Simplify.

$72 - x = \frac{x \cdot 5}{5^{2}}$

Step 2.3.1.4.2

Move $5$ to the left of $x$ .

$72 - x = \frac{5 x}{5^{2}}$

Step 2.3.1.5

Reduce the expression by cancelling the common factors.

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Step 2.3.1.5.1

Raise $5$ to the power of $2$ .

$72 - x = \frac{5 x}{25}$

Step 2.3.1.5.2

Cancel the common factor of $5$ and $25$ .

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Step 2.3.1.5.2.1

Factor $5$ out of $5 x$ .

$72 - x = \frac{5 (x)}{25}$

Step 2.3.1.5.2.2

Cancel the common factors.

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Step 2.3.1.5.2.2.1

Factor $5$ out of $25$ .

$72 - x = \frac{5 x}{5 \cdot 5}$

Step 2.3.1.5.2.2.2

Cancel the common factor.

$72 - x = \frac{5 x}{5 \cdot 5}$

Step 2.3.1.5.2.2.3

Rewrite the expression.

$72 - x = \frac{x}{5}$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $5$ .

$(72 - x) \cdot 5 = \frac{x}{5} \cdot 5$

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

Simplify $(72 - x) \cdot 5$ .

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Step 3.2.1.1.1

Apply the distributive property.

$72 \cdot 5 - x \cdot 5 = \frac{x}{5} \cdot 5$

Step 3.2.1.1.2

Simplify the expression.

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Step 3.2.1.1.2.1

Multiply $72$ by $5$ .

$360 - x \cdot 5 = \frac{x}{5} \cdot 5$

Step 3.2.1.1.2.2

Multiply $5$ by $- 1$ .

$360 - 5 x = \frac{x}{5} \cdot 5$

Step 3.2.1.1.2.3

Reorder $360$ and $- 5 x$ .

$- 5 x + 360 = \frac{x}{5} \cdot 5$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$- 5 x + 360 = \frac{x}{5} \cdot 5$

Step 3.2.2.1.2

Rewrite the expression.

$- 5 x + 360 = x$

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

Subtract $x$ from both sides of the equation.

$- 5 x + 360 - x = 0$

Step 3.3.1.2

Subtract $x$ from $- 5 x$ .

$- 6 x + 360 = 0$

Step 3.3.2

Subtract $360$ from both sides of the equation.

$- 6 x = - 360$

Step 3.3.3

Divide each term in $- 6 x = - 360$ by $- 6$ and simplify.

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Step 3.3.3.1

Divide each term in $- 6 x = - 360$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- 360}{- 6}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 6$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- 360}{- 6}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 360}{- 6}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $- 360$ by $- 6$ .

$x = 60$