Pre-Algebra Examples

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

Step 1

Rewrite the equation as ${(\frac{3 \sqrt{2}}{4})}^{2} + x^{2} = {(\frac{\sqrt{5}}{2})}^{2}$ .

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

Step 2

Simplify each term.

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Step 2.1

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

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Step 2.1.1

Apply the product rule to $\frac{3 \sqrt{2}}{4}$ .

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

Step 2.1.2

Apply the product rule to $3 \sqrt{2}$ .

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

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

$\frac{9 {\sqrt{2}}^{2}}{4^{2}} + x^{2} = {(\frac{\sqrt{5}}{2})}^{2}$

Step 2.2.2

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

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Step 2.2.2.1

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

$\frac{9 {(2^{\frac{1}{2}})}^{2}}{4^{2}} + x^{2} = {(\frac{\sqrt{5}}{2})}^{2}$

Step 2.2.2.2

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

$\frac{9 \cdot 2^{\frac{1}{2} \cdot 2}}{4^{2}} + x^{2} = {(\frac{\sqrt{5}}{2})}^{2}$

Step 2.2.2.3

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

$\frac{9 \cdot 2^{\frac{2}{2}}}{4^{2}} + x^{2} = {(\frac{\sqrt{5}}{2})}^{2}$

Step 2.2.2.4

Cancel the common factor of $2$ .

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Step 2.2.2.4.1

Cancel the common factor.

$\frac{9 \cdot 2^{\frac{2}{2}}}{4^{2}} + x^{2} = {(\frac{\sqrt{5}}{2})}^{2}$

Step 2.2.2.4.2

Rewrite the expression.

$\frac{9 \cdot 2^{1}}{4^{2}} + x^{2} = {(\frac{\sqrt{5}}{2})}^{2}$

Step 2.2.2.5

Evaluate the exponent.

$\frac{9 \cdot 2}{4^{2}} + x^{2} = {(\frac{\sqrt{5}}{2})}^{2}$

Step 2.3

Raise $4$ to the power of $2$ .

$\frac{9 \cdot 2}{16} + x^{2} = {(\frac{\sqrt{5}}{2})}^{2}$

Step 2.4

Multiply $9$ by $2$ .

$\frac{18}{16} + x^{2} = {(\frac{\sqrt{5}}{2})}^{2}$

Step 2.5

Cancel the common factor of $18$ and $16$ .

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Step 2.5.1

Factor $2$ out of $18$ .

$\frac{2 (9)}{16} + x^{2} = {(\frac{\sqrt{5}}{2})}^{2}$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $2$ out of $16$ .

$\frac{2 \cdot 9}{2 \cdot 8} + x^{2} = {(\frac{\sqrt{5}}{2})}^{2}$

Step 2.5.2.2

Cancel the common factor.

$\frac{2 \cdot 9}{2 \cdot 8} + x^{2} = {(\frac{\sqrt{5}}{2})}^{2}$

Step 2.5.2.3

Rewrite the expression.

$\frac{9}{8} + x^{2} = {(\frac{\sqrt{5}}{2})}^{2}$

Step 3

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

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Step 3.1

Apply the product rule to $\frac{\sqrt{5}}{2}$ .

$\frac{9}{8} + x^{2} = \frac{{\sqrt{5}}^{2}}{2^{2}}$

Step 3.2

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

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Step 3.2.1

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

$\frac{9}{8} + x^{2} = \frac{{(5^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 3.2.2

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

$\frac{9}{8} + x^{2} = \frac{5^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 3.2.3

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

$\frac{9}{8} + x^{2} = \frac{5^{\frac{2}{2}}}{2^{2}}$

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Cancel the common factor.

$\frac{9}{8} + x^{2} = \frac{5^{\frac{2}{2}}}{2^{2}}$

Step 3.2.4.2

Rewrite the expression.

$\frac{9}{8} + x^{2} = \frac{5^{1}}{2^{2}}$

Step 3.2.5

Evaluate the exponent.

$\frac{9}{8} + x^{2} = \frac{5}{2^{2}}$

Step 3.3

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

$\frac{9}{8} + x^{2} = \frac{5}{4}$

Step 4

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

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Step 4.1

Subtract $\frac{9}{8}$ from both sides of the equation.

$x^{2} = \frac{5}{4} - \frac{9}{8}$

Step 4.2

To write $\frac{5}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{2} = \frac{5}{4} \cdot \frac{2}{2} - \frac{9}{8}$

Step 4.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{5}{4}$ by $\frac{2}{2}$ .

$x^{2} = \frac{5 \cdot 2}{4 \cdot 2} - \frac{9}{8}$

Step 4.3.2

Multiply $4$ by $2$ .

$x^{2} = \frac{5 \cdot 2}{8} - \frac{9}{8}$

Step 4.4

Combine the numerators over the common denominator.

$x^{2} = \frac{5 \cdot 2 - 9}{8}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $5$ by $2$ .

$x^{2} = \frac{10 - 9}{8}$

Step 4.5.2

Subtract $9$ from $10$ .

$x^{2} = \frac{1}{8}$

Step 5

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

$x = \pm \sqrt{\frac{1}{8}}$

Step 6

Simplify $\pm \sqrt{\frac{1}{8}}$ .

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Step 6.1

Rewrite $\sqrt{\frac{1}{8}}$ as $\frac{\sqrt{1}}{\sqrt{8}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{8}}$

Step 6.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{8}}$

Step 6.3

Simplify the denominator.

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Step 6.3.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 6.3.1.1

Factor $4$ out of $8$ .

$x = \pm \frac{1}{\sqrt{4 (2)}}$

Step 6.3.1.2

Rewrite $4$ as $2^{2}$ .

$x = \pm \frac{1}{\sqrt{2^{2} \cdot 2}}$

Step 6.3.2

Pull terms out from under the radical.

$x = \pm \frac{1}{2 \sqrt{2}}$

Step 6.4

Multiply $\frac{1}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{1}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 6.5

Combine and simplify the denominator.

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Step 6.5.1

Multiply $\frac{1}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 6.5.2

Move $\sqrt{2}$ .

$x = \pm \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 6.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 6.5.4

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 6.5.5

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

$x = \pm \frac{\sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 6.5.6

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 6.5.7

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

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Step 6.5.7.1

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

$x = \pm \frac{\sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 6.5.7.2

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

$x = \pm \frac{\sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 6.5.7.3

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

$x = \pm \frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 6.5.7.4

Cancel the common factor of $2$ .

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Step 6.5.7.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 6.5.7.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{2}}{2 \cdot 2^{1}}$

Step 6.5.7.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{2}}{2 \cdot 2}$

Step 6.6

Multiply $2$ by $2$ .

$x = \pm \frac{\sqrt{2}}{4}$

Step 7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{2}}{4}$

Step 7.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{2}}{4}$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{2}}{4}, - \frac{\sqrt{2}}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{2}}{4}, - \frac{\sqrt{2}}{4}$

Decimal Form:

$x = 0.35355339 \dots, - 0.35355339 \dots$