Finite Math Examples

$r = - \sqrt{\frac{a}{2 p}}$

Step 1

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

$- \sqrt{\frac{a}{2 p}} = r$

Step 2

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

${(- \sqrt{\frac{a}{2 p}})}^{2} = r^{2}$

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 $\sqrt{\frac{a}{2 p}}$ as ${(\frac{a}{2 p})}^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(- {(\frac{a}{2 p})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

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

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Step 3.2.1.1.1

Apply the product rule to $\frac{a}{2 p}$ .

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

Step 3.2.1.1.2

Apply the product rule to $2 p$ .

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

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 $- \frac{a^{\frac{1}{2}}}{2^{\frac{1}{2}} p^{\frac{1}{2}}}$ .

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

Step 3.2.1.2.2

Apply the product rule to $\frac{a^{\frac{1}{2}}}{2^{\frac{1}{2}} p^{\frac{1}{2}}}$ .

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

Step 3.2.1.2.3

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

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

Step 3.2.1.3

Simplify the expression.

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Step 3.2.1.3.1

Raise $- 1$ to the power of $2$ .

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

Step 3.2.1.3.2

Multiply $\frac{{(a^{\frac{1}{2}})}^{2}}{{(2^{\frac{1}{2}})}^{2} {(p^{\frac{1}{2}})}^{2}}$ by $1$ .

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

Step 3.2.1.4

Simplify the numerator.

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Step 3.2.1.4.1

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

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Step 3.2.1.4.1.1

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

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

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.

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

Step 3.2.1.4.1.2.2

Rewrite the expression.

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

Step 3.2.1.4.2

Simplify.

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

Step 3.2.1.5

Simplify the denominator.

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Step 3.2.1.5.1

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

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Step 3.2.1.5.1.1

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

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

Step 3.2.1.5.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.5.1.2.1

Cancel the common factor.

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

Step 3.2.1.5.1.2.2

Rewrite the expression.

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

Step 3.2.1.5.2

Evaluate the exponent.

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

Step 3.2.1.5.3

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

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Step 3.2.1.5.3.1

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

$\frac{a}{2 p^{\frac{1}{2} \cdot 2}} = r^{2}$

Step 3.2.1.5.3.2

Cancel the common factor of $2$ .

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Step 3.2.1.5.3.2.1

Cancel the common factor.

$\frac{a}{2 p^{\frac{1}{2} \cdot 2}} = r^{2}$

Step 3.2.1.5.3.2.2

Rewrite the expression.

$\frac{a}{2 p^{1}} = r^{2}$

Step 3.2.1.5.4

Simplify.

$\frac{a}{2 p} = r^{2}$

Step 4

Solve for $a$ .

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Step 4.1

Multiply both sides by $2 p$ .

$\frac{a}{2 p} (2 p) = r^{2} (2 p)$

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 $\frac{a}{2 p} (2 p)$ .

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Step 4.2.1.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{a}{2 p} p = r^{2} (2 p)$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Factor $2$ out of $2 p$ .

$2 \frac{a}{2 (p)} p = r^{2} (2 p)$

Step 4.2.1.1.2.2

Cancel the common factor.

$2 \frac{a}{2 p} p = r^{2} (2 p)$

Step 4.2.1.1.2.3

Rewrite the expression.

$\frac{a}{p} p = r^{2} (2 p)$

Step 4.2.1.1.3

Cancel the common factor of $p$ .

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Step 4.2.1.1.3.1

Cancel the common factor.

$\frac{a}{p} p = r^{2} (2 p)$

Step 4.2.1.1.3.2

Rewrite the expression.

$a = r^{2} (2 p)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Rewrite using the commutative property of multiplication.

$a = 2 r^{2} p$