Algebra Examples

$a = \frac{h}{2} \sqrt{1 - r}$

Step 1

Rewrite the equation as $\frac{h}{2} \cdot \sqrt{1 - r} = a$ .

$\frac{h}{2} \cdot \sqrt{1 - r} = a$

Step 2

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

${(\frac{h}{2} \cdot \sqrt{1 - r})}^{2} = a^{2}$

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

${(\frac{h}{2} \cdot {(1 - r)}^{\frac{1}{2}})}^{2} = a^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Combine $\frac{h}{2}$ and ${(1 - r)}^{\frac{1}{2}}$ .

${(\frac{h {(1 - r)}^{\frac{1}{2}}}{2})}^{2} = a^{2}$

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

$\frac{{(h {(1 - r)}^{\frac{1}{2}})}^{2}}{2^{2}} = a^{2}$

Step 3.2.1.2.2

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

$\frac{h^{2} {({(1 - r)}^{\frac{1}{2}})}^{2}}{2^{2}} = a^{2}$

Step 3.2.1.3

Simplify the numerator.

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Step 3.2.1.3.1

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

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Step 3.2.1.3.1.1

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

$\frac{h^{2} {(1 - r)}^{\frac{1}{2} \cdot 2}}{2^{2}} = a^{2}$

Step 3.2.1.3.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.3.1.2.1

Cancel the common factor.

$\frac{h^{2} {(1 - r)}^{\frac{1}{2} \cdot 2}}{2^{2}} = a^{2}$

Step 3.2.1.3.1.2.2

Rewrite the expression.

$\frac{h^{2} {(1 - r)}^{1}}{2^{2}} = a^{2}$

Step 3.2.1.3.2

Simplify.

$\frac{h^{2} (1 - r)}{2^{2}} = a^{2}$

Step 3.2.1.4

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

$\frac{h^{2} (1 - r)}{4} = a^{2}$

Step 4

Solve for $r$ .

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Step 4.1

Multiply both sides by $4$ .

$\frac{h^{2} (1 - r)}{4} \cdot 4 = a^{2} \cdot 4$

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $\frac{h^{2} (1 - r)}{4} \cdot 4$ .

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Step 4.2.1.1.1

Cancel the common factor of $4$ .

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Step 4.2.1.1.1.1

Cancel the common factor.

$\frac{h^{2} (1 - r)}{4} \cdot 4 = a^{2} \cdot 4$

Step 4.2.1.1.1.2

Rewrite the expression.

$h^{2} (1 - r) = a^{2} \cdot 4$

Step 4.2.1.1.2

Apply the distributive property.

$h^{2} \cdot 1 + h^{2} (- r) = a^{2} \cdot 4$

Step 4.2.1.1.3

Simplify the expression.

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Step 4.2.1.1.3.1

Multiply $h^{2}$ by $1$ .

$h^{2} + h^{2} (- r) = a^{2} \cdot 4$

Step 4.2.1.1.3.2

Rewrite using the commutative property of multiplication.

$h^{2} - h^{2} r = a^{2} \cdot 4$

Step 4.2.1.1.3.3

Reorder $h^{2}$ and $- h^{2} r$ .

$- h^{2} r + h^{2} = a^{2} \cdot 4$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Move $4$ to the left of $a^{2}$ .

$- h^{2} r + h^{2} = 4 a^{2}$

Step 4.3

Solve for $r$ .

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Step 4.3.1

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

$- h^{2} r = 4 a^{2} - h^{2}$

Step 4.3.2

Divide each term in $- h^{2} r = 4 a^{2} - h^{2}$ by $- h^{2}$ and simplify.

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Step 4.3.2.1

Divide each term in $- h^{2} r = 4 a^{2} - h^{2}$ by $- h^{2}$ .

$\frac{- h^{2} r}{- h^{2}} = \frac{4 a^{2}}{- h^{2}} + \frac{- h^{2}}{- h^{2}}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{h^{2} r}{h^{2}} = \frac{4 a^{2}}{- h^{2}} + \frac{- h^{2}}{- h^{2}}$

Step 4.3.2.2.2

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

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Step 4.3.2.2.2.1

Cancel the common factor.

$\frac{h^{2} r}{h^{2}} = \frac{4 a^{2}}{- h^{2}} + \frac{- h^{2}}{- h^{2}}$

Step 4.3.2.2.2.2

Divide $r$ by $1$ .

$r = \frac{4 a^{2}}{- h^{2}} + \frac{- h^{2}}{- h^{2}}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

Simplify each term.

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Step 4.3.2.3.1.1

Move the negative in front of the fraction.

$r = - \frac{4 a^{2}}{h^{2}} + \frac{- h^{2}}{- h^{2}}$

Step 4.3.2.3.1.2

Dividing two negative values results in a positive value.

$r = - \frac{4 a^{2}}{h^{2}} + \frac{h^{2}}{h^{2}}$

Step 4.3.2.3.1.3

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

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Step 4.3.2.3.1.3.1

Cancel the common factor.

$r = - \frac{4 a^{2}}{h^{2}} + \frac{h^{2}}{h^{2}}$

Step 4.3.2.3.1.3.2

Rewrite the expression.

$r = - \frac{4 a^{2}}{h^{2}} + 1$