Algebra Examples

$d = \frac{5}{6} \sqrt{h}$

Step 1

Rewrite the equation as $\frac{5}{6} \cdot \sqrt{h} = d$ .

$\frac{5}{6} \cdot \sqrt{h} = d$

Step 2

Multiply both sides of the equation by $\frac{6}{5}$ .

$\frac{6}{5} (\frac{5}{6} \cdot \sqrt{h}) = \frac{6}{5} d$

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 $\frac{6}{5} (\frac{5}{6} \cdot \sqrt{h})$ .

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Step 3.1.1.1

Combine $\frac{5}{6}$ and $\sqrt{h}$ .

$\frac{6}{5} \cdot \frac{5 \sqrt{h}}{6} = \frac{6}{5} d$

Step 3.1.1.2

Cancel the common factor of $6$ .

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Step 3.1.1.2.1

Cancel the common factor.

$\frac{6}{5} \cdot \frac{5 \sqrt{h}}{6} = \frac{6}{5} d$

Step 3.1.1.2.2

Rewrite the expression.

$\frac{1}{5} (5 \sqrt{h}) = \frac{6}{5} d$

Step 3.1.1.3

Cancel the common factor of $5$ .

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Step 3.1.1.3.1

Factor $5$ out of $5 \sqrt{h}$ .

$\frac{1}{5} (5 (\sqrt{h})) = \frac{6}{5} d$

Step 3.1.1.3.2

Cancel the common factor.

$\frac{1}{5} (5 \sqrt{h}) = \frac{6}{5} d$

Step 3.1.1.3.3

Rewrite the expression.

$\sqrt{h} = \frac{6}{5} d$

Step 3.2

Simplify the right side.

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Step 3.2.1

Combine $\frac{6}{5}$ and $d$ .

$\sqrt{h} = \frac{6 d}{5}$

Step 4

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

${\sqrt{h}}^{2} = {(\frac{6 d}{5})}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

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

${(h^{\frac{1}{2}})}^{2} = {(\frac{6 d}{5})}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

$h^{\frac{1}{2} \cdot 2} = {(\frac{6 d}{5})}^{2}$

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

$h^{\frac{1}{2} \cdot 2} = {(\frac{6 d}{5})}^{2}$

Step 5.2.1.1.2.2

Rewrite the expression.

$h^{1} = {(\frac{6 d}{5})}^{2}$

Step 5.2.1.2

Simplify.

$h = {(\frac{6 d}{5})}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify ${(\frac{6 d}{5})}^{2}$ .

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Step 5.3.1.1

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

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Step 5.3.1.1.1

Apply the product rule to $\frac{6 d}{5}$ .

$h = \frac{{(6 d)}^{2}}{5^{2}}$

Step 5.3.1.1.2

Apply the product rule to $6 d$ .

$h = \frac{6^{2} d^{2}}{5^{2}}$

Step 5.3.1.2

Raise $6$ to the power of $2$ .

$h = \frac{36 d^{2}}{5^{2}}$

Step 5.3.1.3

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

$h = \frac{36 d^{2}}{25}$

Step 6

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

$f (d) = \frac{36 d^{2}}{25}$