Precalculus Examples

$4 \sqrt{3 m^{2} - 15} = 4$

Step 1

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

${(4 \sqrt{3 m^{2} - 15})}^{2} = 4^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 m^{2} - 15}$ as ${(3 m^{2} - 15)}^{\frac{1}{2}}$ .

${(4 {(3 m^{2} - 15)}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(4 {(3 m^{2} - 15)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Apply the product rule to $4 {(3 m^{2} - 15)}^{\frac{1}{2}}$ .

$4^{2} {({(3 m^{2} - 15)}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 2.2.1.2

Raise $4$ to the power of $2$ .

$16 {({(3 m^{2} - 15)}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 2.2.1.3

Multiply the exponents in ${({(3 m^{2} - 15)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.3.1

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

$16 {(3 m^{2} - 15)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 2.2.1.3.2

Cancel the common factor of $2$ .

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Step 2.2.1.3.2.1

Cancel the common factor.

$16 {(3 m^{2} - 15)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 2.2.1.3.2.2

Rewrite the expression.

$16 {(3 m^{2} - 15)}^{1} = 4^{2}$

Step 2.2.1.4

Simplify.

$16 (3 m^{2} - 15) = 4^{2}$

Step 2.2.1.5

Apply the distributive property.

$16 (3 m^{2}) + 16 \cdot - 15 = 4^{2}$

Step 2.2.1.6

Multiply.

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Step 2.2.1.6.1

Multiply $3$ by $16$ .

$48 m^{2} + 16 \cdot - 15 = 4^{2}$

Step 2.2.1.6.2

Multiply $16$ by $- 15$ .

$48 m^{2} - 240 = 4^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $4$ to the power of $2$ .

$48 m^{2} - 240 = 16$

Step 3

Solve for $m$ .

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Step 3.1

Move all terms not containing $m$ to the right side of the equation.

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Step 3.1.1

Add $240$ to both sides of the equation.

$48 m^{2} = 16 + 240$

Step 3.1.2

Add $16$ and $240$ .

$48 m^{2} = 256$

Step 3.2

Divide each term in $48 m^{2} = 256$ by $48$ and simplify.

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Step 3.2.1

Divide each term in $48 m^{2} = 256$ by $48$ .

$\frac{48 m^{2}}{48} = \frac{256}{48}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $48$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{48 m^{2}}{48} = \frac{256}{48}$

Step 3.2.2.1.2

Divide $m^{2}$ by $1$ .

$m^{2} = \frac{256}{48}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $256$ and $48$ .

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Step 3.2.3.1.1

Factor $16$ out of $256$ .

$m^{2} = \frac{16 (16)}{48}$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $16$ out of $48$ .

$m^{2} = \frac{16 \cdot 16}{16 \cdot 3}$

Step 3.2.3.1.2.2

Cancel the common factor.

$m^{2} = \frac{16 \cdot 16}{16 \cdot 3}$

Step 3.2.3.1.2.3

Rewrite the expression.

$m^{2} = \frac{16}{3}$

Step 3.3

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

$m = \pm \sqrt{\frac{16}{3}}$

Step 3.4

Simplify $\pm \sqrt{\frac{16}{3}}$ .

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Step 3.4.1

Rewrite $\sqrt{\frac{16}{3}}$ as $\frac{\sqrt{16}}{\sqrt{3}}$ .

$m = \pm \frac{\sqrt{16}}{\sqrt{3}}$

Step 3.4.2

Simplify the numerator.

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Step 3.4.2.1

Rewrite $16$ as $4^{2}$ .

$m = \pm \frac{\sqrt{4^{2}}}{\sqrt{3}}$

Step 3.4.2.2

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

$m = \pm \frac{4}{\sqrt{3}}$

Step 3.4.3

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$m = \pm \frac{4}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 3.4.4

Combine and simplify the denominator.

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Step 3.4.4.1

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$m = \pm \frac{4 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.4.4.2

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

$m = \pm \frac{4 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.4.4.3

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

$m = \pm \frac{4 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.4.4.4

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

$m = \pm \frac{4 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.4.4.5

Add $1$ and $1$ .

$m = \pm \frac{4 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.4.4.6

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

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Step 3.4.4.6.1

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

$m = \pm \frac{4 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.4.4.6.2

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

$m = \pm \frac{4 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.4.4.6.3

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

$m = \pm \frac{4 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.4.4.6.4

Cancel the common factor of $2$ .

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Step 3.4.4.6.4.1

Cancel the common factor.

$m = \pm \frac{4 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.4.4.6.4.2

Rewrite the expression.

$m = \pm \frac{4 \sqrt{3}}{3^{1}}$

Step 3.4.4.6.5

Evaluate the exponent.

$m = \pm \frac{4 \sqrt{3}}{3}$

Step 3.5

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

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Step 3.5.1

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

$m = \frac{4 \sqrt{3}}{3}$

Step 3.5.2

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

$m = - \frac{4 \sqrt{3}}{3}$

Step 3.5.3

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

$m = \frac{4 \sqrt{3}}{3}, - \frac{4 \sqrt{3}}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$m = \frac{4 \sqrt{3}}{3}, - \frac{4 \sqrt{3}}{3}$

Decimal Form:

$m = 2.30940107 \dots, - 2.30940107 \dots$