Algebra Examples

$v = \sqrt{2 g r} \cdot \sqrt{\frac{h}{r + h}}$

Step 1

Rewrite the equation as $\sqrt{2 g r} \cdot \sqrt{\frac{h}{r + h}} = v$ .

$\sqrt{2 g r} \cdot \sqrt{\frac{h}{r + h}} = v$

Step 2

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

${(\sqrt{2 g r} \cdot \sqrt{\frac{h}{r + h}})}^{2} = v^{2}$

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 $\sqrt{\frac{h}{r + h}}$ as ${(\frac{h}{r + h})}^{\frac{1}{2}}$ .

${(\sqrt{2 g r} \cdot {(\frac{h}{r + h})}^{\frac{1}{2}})}^{2} = v^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(\sqrt{2 g r} \cdot {(\frac{h}{r + h})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $\frac{h}{r + h}$ .

${(\frac{\sqrt{2 g r} h^{\frac{1}{2}}}{{(r + h)}^{\frac{1}{2}}})}^{2} = v^{2}$

Step 3.2.1.2

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

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Step 3.2.1.2.1

Apply the product rule to $\frac{\sqrt{2 g r} h^{\frac{1}{2}}}{{(r + h)}^{\frac{1}{2}}}$ .

$\frac{{(\sqrt{2 g r} h^{\frac{1}{2}})}^{2}}{{({(r + h)}^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.2.2

Apply the product rule to $\sqrt{2 g r} h^{\frac{1}{2}}$ .

$\frac{{\sqrt{2 g r}}^{2} {(h^{\frac{1}{2}})}^{2}}{{({(r + h)}^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.3

Simplify the numerator.

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Step 3.2.1.3.1

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

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Step 3.2.1.3.1.1

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

$\frac{{({(2 g r)}^{\frac{1}{2}})}^{2} {(h^{\frac{1}{2}})}^{2}}{{({(r + h)}^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.3.1.2

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

$\frac{{(2 g r)}^{\frac{1}{2} \cdot 2} {(h^{\frac{1}{2}})}^{2}}{{({(r + h)}^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.3.1.3

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

$\frac{{(2 g r)}^{\frac{2}{2}} {(h^{\frac{1}{2}})}^{2}}{{({(r + h)}^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.3.1.4

Cancel the common factor of $2$ .

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Step 3.2.1.3.1.4.1

Cancel the common factor.

$\frac{{(2 g r)}^{\frac{2}{2}} {(h^{\frac{1}{2}})}^{2}}{{({(r + h)}^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.3.1.4.2

Rewrite the expression.

$\frac{{(2 g r)}^{1} {(h^{\frac{1}{2}})}^{2}}{{({(r + h)}^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.3.1.5

Simplify.

$\frac{2 g r {(h^{\frac{1}{2}})}^{2}}{{({(r + h)}^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.3.2

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

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Step 3.2.1.3.2.1

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

$\frac{2 g r h^{\frac{1}{2} \cdot 2}}{{({(r + h)}^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.3.2.2

Cancel the common factor of $2$ .

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Step 3.2.1.3.2.2.1

Cancel the common factor.

$\frac{2 g r h^{\frac{1}{2} \cdot 2}}{{({(r + h)}^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.3.2.2.2

Rewrite the expression.

$\frac{2 g r h^{1}}{{({(r + h)}^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.3.3

Simplify.

$\frac{2 g r h}{{({(r + h)}^{\frac{1}{2}})}^{2}} = v^{2}$

Step 3.2.1.4

Simplify the denominator.

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Step 3.2.1.4.1

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

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Step 3.2.1.4.1.1

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

$\frac{2 g r h}{{(r + h)}^{\frac{1}{2} \cdot 2}} = v^{2}$

Step 3.2.1.4.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.4.1.2.1

Cancel the common factor.

$\frac{2 g r h}{{(r + h)}^{\frac{1}{2} \cdot 2}} = v^{2}$

Step 3.2.1.4.1.2.2

Rewrite the expression.

$\frac{2 g r h}{{(r + h)}^{1}} = v^{2}$

Step 3.2.1.4.2

Simplify.

$\frac{2 g r h}{r + h} = v^{2}$

Step 4

Solve for $h$ .

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Step 4.1

Find the LCD of the terms in the equation.

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Step 4.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$r + h, 1$

Step 4.1.2

Remove parentheses.

$r + h, 1$

Step 4.1.3

The LCM of one and any expression is the expression.

$r + h$

Step 4.2

Multiply each term in $\frac{2 g r h}{r + h} = v^{2}$ by $r + h$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $\frac{2 g r h}{r + h} = v^{2}$ by $r + h$ .

$\frac{2 g r h}{r + h} (r + h) = v^{2} (r + h)$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $r + h$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{2 g r h}{r + h} (r + h) = v^{2} (r + h)$

Step 4.2.2.1.2

Rewrite the expression.

$2 g r h = v^{2} (r + h)$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Apply the distributive property.

$2 g r h = v^{2} r + v^{2} h$

Step 4.3

Solve the equation.

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Step 4.3.1

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

$2 g r h - v^{2} h = v^{2} r$

Step 4.3.2

Factor $h$ out of $2 g r h - v^{2} h$ .

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Step 4.3.2.1

Factor $h$ out of $2 g r h$ .

$h (2 g r) - v^{2} h = v^{2} r$

Step 4.3.2.2

Factor $h$ out of $- v^{2} h$ .

$h (2 g r) + h (- v^{2}) = v^{2} r$

Step 4.3.2.3

Factor $h$ out of $h (2 g r) + h (- v^{2})$ .

$h (2 g r - v^{2}) = v^{2} r$

Step 4.3.3

Divide each term in $h (2 g r - v^{2}) = v^{2} r$ by $2 g r - v^{2}$ and simplify.

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Step 4.3.3.1

Divide each term in $h (2 g r - v^{2}) = v^{2} r$ by $2 g r - v^{2}$ .

$\frac{h (2 g r - v^{2})}{2 g r - v^{2}} = \frac{v^{2} r}{2 g r - v^{2}}$

Step 4.3.3.2

Simplify the left side.

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Step 4.3.3.2.1

Cancel the common factor of $2 g r - v^{2}$ .

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Step 4.3.3.2.1.1

Cancel the common factor.

$\frac{h (2 g r - v^{2})}{2 g r - v^{2}} = \frac{v^{2} r}{2 g r - v^{2}}$

Step 4.3.3.2.1.2

Divide $h$ by $1$ .

$h = \frac{v^{2} r}{2 g r - v^{2}}$