Finite Math Examples

$p^{2} = \frac{75}{81}$

Step 1

Reduce the expression $\frac{75}{81}$ by cancelling the common factors.

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Step 1.1

Factor $3$ out of $75$ .

$p^{2} = \frac{3 (25)}{81}$

Step 1.2

Factor $3$ out of $81$ .

$p^{2} = \frac{3 \cdot 25}{3 \cdot 27}$

Step 1.3

Cancel the common factor.

$p^{2} = \frac{3 \cdot 25}{3 \cdot 27}$

Step 1.4

Rewrite the expression.

$p^{2} = \frac{25}{27}$

Step 2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$p = \pm \sqrt{\frac{25}{27}}$

Step 3

Simplify $\pm \sqrt{\frac{25}{27}}$ .

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Step 3.1

Rewrite $\sqrt{\frac{25}{27}}$ as $\frac{\sqrt{25}}{\sqrt{27}}$ .

$p = \pm \frac{\sqrt{25}}{\sqrt{27}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Rewrite $25$ as $5^{2}$ .

$p = \pm \frac{\sqrt{5^{2}}}{\sqrt{27}}$

Step 3.2.2

Pull terms out from under the radical, assuming positive real numbers.

$p = \pm \frac{5}{\sqrt{27}}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 3.3.1.1

Factor $9$ out of $27$ .

$p = \pm \frac{5}{\sqrt{9 (3)}}$

Step 3.3.1.2

Rewrite $9$ as $3^{2}$ .

$p = \pm \frac{5}{\sqrt{3^{2} \cdot 3}}$

Step 3.3.2

Pull terms out from under the radical.

$p = \pm \frac{5}{3 \sqrt{3}}$

Step 3.4

Multiply $\frac{5}{3 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$p = \pm \frac{5}{3 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

Multiply $\frac{5}{3 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$p = \pm \frac{5 \sqrt{3}}{3 \sqrt{3} \sqrt{3}}$

Step 3.5.2

Move $\sqrt{3}$ .

$p = \pm \frac{5 \sqrt{3}}{3 (\sqrt{3} \sqrt{3})}$

Step 3.5.3

Raise $\sqrt{3}$ to the power of $1$ .

$p = \pm \frac{5 \sqrt{3}}{3 ({\sqrt{3}}^{1} \sqrt{3})}$

Step 3.5.4

Raise $\sqrt{3}$ to the power of $1$ .

$p = \pm \frac{5 \sqrt{3}}{3 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Step 3.5.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p = \pm \frac{5 \sqrt{3}}{3 {\sqrt{3}}^{1 + 1}}$

Step 3.5.6

Add $1$ and $1$ .

$p = \pm \frac{5 \sqrt{3}}{3 {\sqrt{3}}^{2}}$

Step 3.5.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$p = \pm \frac{5 \sqrt{3}}{3 {(3^{\frac{1}{2}})}^{2}}$

Step 3.5.7.2

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

$p = \pm \frac{5 \sqrt{3}}{3 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 3.5.7.3

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

$p = \pm \frac{5 \sqrt{3}}{3 \cdot 3^{\frac{2}{2}}}$

Step 3.5.7.4

Cancel the common factor of $2$ .

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Step 3.5.7.4.1

Cancel the common factor.

$p = \pm \frac{5 \sqrt{3}}{3 \cdot 3^{\frac{2}{2}}}$

Step 3.5.7.4.2

Rewrite the expression.

$p = \pm \frac{5 \sqrt{3}}{3 \cdot 3^{1}}$

Step 3.5.7.5

Evaluate the exponent.

$p = \pm \frac{5 \sqrt{3}}{3 \cdot 3}$

Step 3.6

Multiply $3$ by $3$ .

$p = \pm \frac{5 \sqrt{3}}{9}$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$p = \frac{5 \sqrt{3}}{9}$

Step 4.2

Next, use the negative value of the $\pm$ to find the second solution.

$p = - \frac{5 \sqrt{3}}{9}$

Step 4.3

The complete solution is the result of both the positive and negative portions of the solution.

$p = \frac{5 \sqrt{3}}{9}, - \frac{5 \sqrt{3}}{9}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$p = \frac{5 \sqrt{3}}{9}, - \frac{5 \sqrt{3}}{9}$

Decimal Form:

$p = 0.96225044 \dots, - 0.96225044 \dots$