Finite Math Examples

$\sqrt{\frac{x - 8 + 4}{2}} + \sqrt{81 \cdot 4} = 3^{2} (\frac{3}{5})$

Step 1

Solve for $\sqrt{\frac{x - 8 + 4}{2}}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Multiply $81$ by $4$ .

$\sqrt{\frac{x - 8 + 4}{2}} + \sqrt{324} = 3^{2} (\frac{3}{5})$

Step 1.1.2

Rewrite $324$ as $18^{2}$ .

$\sqrt{\frac{x - 8 + 4}{2}} + \sqrt{18^{2}} = 3^{2} (\frac{3}{5})$

Step 1.1.3

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

$\sqrt{\frac{x - 8 + 4}{2}} + 18 = 3^{2} (\frac{3}{5})$

Step 1.2

Simplify $3^{2} (\frac{3}{5})$ .

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Step 1.2.1

Raise $3$ to the power of $2$ .

$\sqrt{\frac{x - 8 + 4}{2}} + 18 = 9 (\frac{3}{5})$

Step 1.2.2

Multiply $9 (\frac{3}{5})$ .

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Step 1.2.2.1

Combine $9$ and $\frac{3}{5}$ .

$\sqrt{\frac{x - 8 + 4}{2}} + 18 = \frac{9 \cdot 3}{5}$

Step 1.2.2.2

Multiply $9$ by $3$ .

$\sqrt{\frac{x - 8 + 4}{2}} + 18 = \frac{27}{5}$

Step 1.3

Move all terms not containing $\sqrt{\frac{x - 8 + 4}{2}}$ to the right side of the equation.

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Step 1.3.1

Subtract $18$ from both sides of the equation.

$\sqrt{\frac{x - 8 + 4}{2}} = \frac{27}{5} - 18$

Step 1.3.2

To write $- 18$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\sqrt{\frac{x - 8 + 4}{2}} = \frac{27}{5} - 18 \cdot \frac{5}{5}$

Step 1.3.3

Combine $- 18$ and $\frac{5}{5}$ .

$\sqrt{\frac{x - 8 + 4}{2}} = \frac{27}{5} + \frac{- 18 \cdot 5}{5}$

Step 1.3.4

Combine the numerators over the common denominator.

$\sqrt{\frac{x - 8 + 4}{2}} = \frac{27 - 18 \cdot 5}{5}$

Step 1.3.5

Simplify the numerator.

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Step 1.3.5.1

Multiply $- 18$ by $5$ .

$\sqrt{\frac{x - 8 + 4}{2}} = \frac{27 - 90}{5}$

Step 1.3.5.2

Subtract $90$ from $27$ .

$\sqrt{\frac{x - 8 + 4}{2}} = \frac{- 63}{5}$

Step 1.3.6

Move the negative in front of the fraction.

$\sqrt{\frac{x - 8 + 4}{2}} = - \frac{63}{5}$

Step 2

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

${\sqrt{\frac{x - 8 + 4}{2}}}^{2} = {(- \frac{63}{5})}^{2}$

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 $\sqrt{\frac{x - 8 + 4}{2}}$ as ${(\frac{x - 8 + 4}{2})}^{\frac{1}{2}}$ .

${({(\frac{x - 8 + 4}{2})}^{\frac{1}{2}})}^{2} = {(- \frac{63}{5})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(\frac{x - 8 + 4}{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(\frac{x - 8 + 4}{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(\frac{x - 8 + 4}{2})}^{\frac{1}{2} \cdot 2} = {(- \frac{63}{5})}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(\frac{x - 8 + 4}{2})}^{\frac{1}{2} \cdot 2} = {(- \frac{63}{5})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(\frac{x - 8 + 4}{2})}^{1} = {(- \frac{63}{5})}^{2}$

Step 3.2.1.2

Add $- 8$ and $4$ .

${(\frac{x - 4}{2})}^{1} = {(- \frac{63}{5})}^{2}$

Step 3.2.1.3

Simplify.

$\frac{x - 4}{2} = {(- \frac{63}{5})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(- \frac{63}{5})}^{2}$ .

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Step 3.3.1.1

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

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Step 3.3.1.1.1

Apply the product rule to $- \frac{63}{5}$ .

$\frac{x - 4}{2} = {(- 1)}^{2} {(\frac{63}{5})}^{2}$

Step 3.3.1.1.2

Apply the product rule to $\frac{63}{5}$ .

$\frac{x - 4}{2} = {(- 1)}^{2} \frac{63^{2}}{5^{2}}$

Step 3.3.1.2

Raise $- 1$ to the power of $2$ .

$\frac{x - 4}{2} = 1 \frac{63^{2}}{5^{2}}$

Step 3.3.1.3

Multiply $\frac{63^{2}}{5^{2}}$ by $1$ .

$\frac{x - 4}{2} = \frac{63^{2}}{5^{2}}$

Step 3.3.1.4

Raise $63$ to the power of $2$ .

$\frac{x - 4}{2} = \frac{3969}{5^{2}}$

Step 3.3.1.5

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

$\frac{x - 4}{2} = \frac{3969}{25}$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides of the equation by $2$ .

$2 \frac{x - 4}{2} = 2 (\frac{3969}{25})$

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1

Cancel the common factor.

$2 \frac{x - 4}{2} = 2 (\frac{3969}{25})$

Step 4.2.1.1.2

Rewrite the expression.

$x - 4 = 2 (\frac{3969}{25})$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $2 (\frac{3969}{25})$ .

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Step 4.2.2.1.1

Combine $2$ and $\frac{3969}{25}$ .

$x - 4 = \frac{2 \cdot 3969}{25}$

Step 4.2.2.1.2

Multiply $2$ by $3969$ .

$x - 4 = \frac{7938}{25}$

Step 4.3

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

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Step 4.3.1

Add $4$ to both sides of the equation.

$x = \frac{7938}{25} + 4$

Step 4.3.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$x = \frac{7938}{25} + 4 \cdot \frac{25}{25}$

Step 4.3.3

Combine $4$ and $\frac{25}{25}$ .

$x = \frac{7938}{25} + \frac{4 \cdot 25}{25}$

Step 4.3.4

Combine the numerators over the common denominator.

$x = \frac{7938 + 4 \cdot 25}{25}$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $4$ by $25$ .

$x = \frac{7938 + 100}{25}$

Step 4.3.5.2

Add $7938$ and $100$ .

$x = \frac{8038}{25}$

Step 5

Exclude the solutions that do not make $\sqrt{\frac{x - 8 + 4}{2}} + \sqrt{81 \cdot 4} = 3^{2} (\frac{3}{5})$ true.

$No solution$