Algebra Examples

$D (h) = 111.7 \sqrt{h}$

Step 1

Rewrite the equation as $111.7 \sqrt{h} = D h$ .

$111.7 \sqrt{h} = D h$

Step 2

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

${(111.7 \sqrt{h})}^{2} = {(D h)}^{2}$

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 $\sqrt{h}$ as $h^{\frac{1}{2}}$ .

${(111.7 h^{\frac{1}{2}})}^{2} = {(D h)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(111.7 h^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $111.7 h^{\frac{1}{2}}$ .

${111.7}^{2} {(h^{\frac{1}{2}})}^{2} = {(D h)}^{2}$

Step 3.2.1.2

Raise $111.7$ to the power of $2$ .

$12476.89 {(h^{\frac{1}{2}})}^{2} = {(D h)}^{2}$

Step 3.2.1.3

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

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Step 3.2.1.3.1

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

$12476.89 h^{\frac{1}{2} \cdot 2} = {(D h)}^{2}$

Step 3.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

$12476.89 h^{\frac{1}{2} \cdot 2} = {(D h)}^{2}$

Step 3.2.1.3.2.2

Rewrite the expression.

$12476.89 h^{1} = {(D h)}^{2}$

Step 3.2.1.4

Simplify.

$12476.89 h = {(D h)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the product rule to $D h$ .

$12476.89 h = D^{2} h^{2}$

Step 4

Solve for $h$ .

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Step 4.1

Subtract $D^{2} h^{2}$ from both sides of the equation.

$12476.89 h - D^{2} h^{2} = 0$

Step 4.2

Factor $0.01 h$ out of $12476.89 h - D^{2} h^{2}$ .

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Step 4.2.1

Factor $0.01 h$ out of $12476.89 h$ .

$0.01 h (1247689) - D^{2} h^{2} = 0$

Step 4.2.2

Factor $0.01 h$ out of $- D^{2} h^{2}$ .

$0.01 h (1247689) + 0.01 h (- 100 D^{2} h) = 0$

Step 4.2.3

Factor $0.01 h$ out of $0.01 h (1247689) + 0.01 h (- 100 D^{2} h)$ .

$0.01 h (1247689 - 100 D^{2} h) = 0$

Step 4.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$h = 0$

$1247689 - 100 D^{2} h = 0$

Step 4.4

Set $h$ equal to $0$ .

$h = 0$

Step 4.5

Set $1247689 - 100 D^{2} h$ equal to $0$ and solve for $h$ .

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Step 4.5.1

Set $1247689 - 100 D^{2} h$ equal to $0$ .

$1247689 - 100 D^{2} h = 0$

Step 4.5.2

Solve $1247689 - 100 D^{2} h = 0$ for $h$ .

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Step 4.5.2.1

Subtract $1247689$ from both sides of the equation.

$- 100 D^{2} h = - 1247689$

Step 4.5.2.2

Divide each term in $- 100 D^{2} h = - 1247689$ by $- 100 D^{2}$ and simplify.

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Step 4.5.2.2.1

Divide each term in $- 100 D^{2} h = - 1247689$ by $- 100 D^{2}$ .

$\frac{- 100 D^{2} h}{- 100 D^{2}} = \frac{- 1247689}{- 100 D^{2}}$

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Cancel the common factor of $- 100$ .

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Step 4.5.2.2.2.1.1

Cancel the common factor.

$\frac{- 100 D^{2} h}{- 100 D^{2}} = \frac{- 1247689}{- 100 D^{2}}$

Step 4.5.2.2.2.1.2

Rewrite the expression.

$\frac{D^{2} h}{D^{2}} = \frac{- 1247689}{- 100 D^{2}}$

Step 4.5.2.2.2.2

Cancel the common factor of $D^{2}$ .

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Step 4.5.2.2.2.2.1

Cancel the common factor.

$\frac{D^{2} h}{D^{2}} = \frac{- 1247689}{- 100 D^{2}}$

Step 4.5.2.2.2.2.2

Divide $h$ by $1$ .

$h = \frac{- 1247689}{- 100 D^{2}}$

Step 4.5.2.2.3

Simplify the right side.

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Step 4.5.2.2.3.1

Dividing two negative values results in a positive value.

$h = \frac{1247689}{100 D^{2}}$

Step 4.6

The final solution is all the values that make $0.01 h (1247689 - 100 D^{2} h) = 0$ true.

$h = 0$

$h = \frac{1247689}{100 D^{2}}$

$h = 0$

$h = \frac{1247689}{100 D^{2}}$