Algebra Examples

$a = \frac{\sqrt{3}}{4} {(s)}^{2}$

Step 1

Rewrite the equation as $\frac{\sqrt{3}}{4} s^{2} = a$ .

$\frac{\sqrt{3}}{4} s^{2} = a$

Step 2

Multiply both sides of the equation by $\frac{4}{\sqrt{3}}$ .

$\frac{4}{\sqrt{3}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{4}{\sqrt{3}} (\frac{\sqrt{3}}{4} s^{2})$ .

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Step 3.1.1.1

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{4}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Step 3.1.1.2

Combine and simplify the denominator.

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Step 3.1.1.2.1

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{4 \sqrt{3}}{\sqrt{3} \sqrt{3}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Step 3.1.1.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{4 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Step 3.1.1.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{4 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Step 3.1.1.2.4

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

$\frac{4 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Step 3.1.1.2.5

Add $1$ and $1$ .

$\frac{4 \sqrt{3}}{{\sqrt{3}}^{2}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Step 3.1.1.2.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.1.1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{4 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Step 3.1.1.2.6.2

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

$\frac{4 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Step 3.1.1.2.6.3

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

$\frac{4 \sqrt{3}}{3^{\frac{2}{2}}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Step 3.1.1.2.6.4

Cancel the common factor of $2$ .

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Step 3.1.1.2.6.4.1

Cancel the common factor.

$\frac{4 \sqrt{3}}{3^{\frac{2}{2}}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Step 3.1.1.2.6.4.2

Rewrite the expression.

$\frac{4 \sqrt{3}}{3^{1}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Step 3.1.1.2.6.5

Evaluate the exponent.

$\frac{4 \sqrt{3}}{3} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Step 3.1.1.3

Combine $\frac{\sqrt{3}}{4}$ and $s^{2}$ .

$\frac{4 \sqrt{3}}{3} \cdot \frac{\sqrt{3} s^{2}}{4} = \frac{4}{\sqrt{3}} a$

Step 3.1.1.4

Combine.

$\frac{4 \sqrt{3} (\sqrt{3} s^{2})}{3 \cdot 4} = \frac{4}{\sqrt{3}} a$

Step 3.1.1.5

Cancel the common factor of $4$ .

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Step 3.1.1.5.1

Cancel the common factor.

$\frac{4 \sqrt{3} (\sqrt{3} s^{2})}{3 \cdot 4} = \frac{4}{\sqrt{3}} a$

Step 3.1.1.5.2

Rewrite the expression.

$\frac{\sqrt{3} (\sqrt{3} s^{2})}{3} = \frac{4}{\sqrt{3}} a$

Step 3.1.1.6

Multiply $\sqrt{3}$ by $\sqrt{3}$ .

$\frac{{\sqrt{3}}^{2} s^{2}}{3} = \frac{4}{\sqrt{3}} a$

Step 3.1.1.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.1.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{{(3^{\frac{1}{2}})}^{2} s^{2}}{3} = \frac{4}{\sqrt{3}} a$

Step 3.1.1.7.2

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

$\frac{3^{\frac{1}{2} \cdot 2} s^{2}}{3} = \frac{4}{\sqrt{3}} a$

Step 3.1.1.7.3

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

$\frac{3^{\frac{2}{2}} s^{2}}{3} = \frac{4}{\sqrt{3}} a$

Step 3.1.1.7.4

Cancel the common factor of $2$ .

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Step 3.1.1.7.4.1

Cancel the common factor.

$\frac{3^{\frac{2}{2}} s^{2}}{3} = \frac{4}{\sqrt{3}} a$

Step 3.1.1.7.4.2

Rewrite the expression.

$\frac{3^{1} s^{2}}{3} = \frac{4}{\sqrt{3}} a$

Step 3.1.1.7.5

Evaluate the exponent.

$\frac{3 s^{2}}{3} = \frac{4}{\sqrt{3}} a$

Step 3.1.1.8

Cancel the common factor of $3$ .

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Step 3.1.1.8.1

Cancel the common factor.

$\frac{3 s^{2}}{3} = \frac{4}{\sqrt{3}} a$

Step 3.1.1.8.2

Divide $s^{2}$ by $1$ .

$s^{2} = \frac{4}{\sqrt{3}} a$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{4}{\sqrt{3}} a$ .

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Step 3.2.1.1

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$s^{2} = \frac{4}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} a$

Step 3.2.1.2

Combine and simplify the denominator.

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Step 3.2.1.2.1

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$s^{2} = \frac{4 \sqrt{3}}{\sqrt{3} \sqrt{3}} a$

Step 3.2.1.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$s^{2} = \frac{4 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} a$

Step 3.2.1.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$s^{2} = \frac{4 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} a$

Step 3.2.1.2.4

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

$s^{2} = \frac{4 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} a$

Step 3.2.1.2.5

Add $1$ and $1$ .

$s^{2} = \frac{4 \sqrt{3}}{{\sqrt{3}}^{2}} a$

Step 3.2.1.2.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.2.1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$s^{2} = \frac{4 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} a$

Step 3.2.1.2.6.2

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

$s^{2} = \frac{4 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} a$

Step 3.2.1.2.6.3

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

$s^{2} = \frac{4 \sqrt{3}}{3^{\frac{2}{2}}} a$

Step 3.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 3.2.1.2.6.4.1

Cancel the common factor.

$s^{2} = \frac{4 \sqrt{3}}{3^{\frac{2}{2}}} a$

Step 3.2.1.2.6.4.2

Rewrite the expression.

$s^{2} = \frac{4 \sqrt{3}}{3^{1}} a$

Step 3.2.1.2.6.5

Evaluate the exponent.

$s^{2} = \frac{4 \sqrt{3}}{3} a$

Step 3.2.1.3

Combine $\frac{4 \sqrt{3}}{3}$ and $a$ .

$s^{2} = \frac{4 \sqrt{3} a}{3}$

Step 4

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

$s = \pm \sqrt{\frac{4 \sqrt{3} a}{3}}$

Step 5

Simplify $\pm \sqrt{\frac{4 \sqrt{3} a}{3}}$ .

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Step 5.1

Rewrite $\sqrt{\frac{4 \sqrt{3} a}{3}}$ as $\frac{\sqrt{4 \sqrt{3} a}}{\sqrt{3}}$ .

$s = \pm \frac{\sqrt{4 \sqrt{3} a}}{\sqrt{3}}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Rewrite $4 \sqrt{3} a$ as $2^{2} (\sqrt{3} a)$ .

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Step 5.2.1.1

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

$s = \pm \frac{\sqrt{2^{2} \sqrt{3} a}}{\sqrt{3}}$

Step 5.2.1.2

Add parentheses.

$s = \pm \frac{\sqrt{2^{2} (\sqrt{3} a)}}{\sqrt{3}}$

Step 5.2.2

Pull terms out from under the radical.

$s = \pm \frac{2 \sqrt{\sqrt{3} a}}{\sqrt{3}}$

Step 5.3

Multiply $\frac{2 \sqrt{\sqrt{3} a}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$s = \pm \frac{2 \sqrt{\sqrt{3} a}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 5.4

Combine and simplify the denominator.

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Step 5.4.1

Multiply $\frac{2 \sqrt{\sqrt{3} a}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 5.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 5.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 5.4.4

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

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 5.4.5

Add $1$ and $1$ .

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 5.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 5.4.6.2

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

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 5.4.6.3

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

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.4.6.4

Cancel the common factor of $2$ .

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Step 5.4.6.4.1

Cancel the common factor.

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.4.6.4.2

Rewrite the expression.

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{3^{1}}$

Step 5.4.6.5

Evaluate the exponent.

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{3}$

Step 5.5

Combine using the product rule for radicals.

$s = \pm \frac{2 \sqrt{3 \sqrt{3} a}}{3}$

Step 6

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

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Step 6.1

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

$s = \frac{2 \sqrt{3 \sqrt{3} a}}{3}$

Step 6.2

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

$s = - \frac{2 \sqrt{3 \sqrt{3} a}}{3}$

Step 6.3

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

$s = \frac{2 \sqrt{3 \sqrt{3} a}}{3}$

$s = - \frac{2 \sqrt{3 \sqrt{3} a}}{3}$

$s = \frac{2 \sqrt{3 \sqrt{3} a}}{3}$

$s = - \frac{2 \sqrt{3 \sqrt{3} a}}{3}$

Step 7

To rewrite as a function of $a$ , write the equation so that $s$ is by itself on one side of the equal sign and an expression involving only $a$ is on the other side.

$f (a) = \frac{2 \sqrt{3 \sqrt{3} a}}{3}, - \frac{2 \sqrt{3 \sqrt{3} a}}{3}$