Calculus Examples

$\sqrt{2 m^{2} - 10} - 4 = 4$

Step 1

Factor $2$ out of $2 m^{2} - 10$ .

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Step 1.1

Factor $2$ out of $2 m^{2}$ .

$\sqrt{2 (m^{2}) - 10} - 4 = 4$

Step 1.2

Factor $2$ out of $- 10$ .

$\sqrt{2 m^{2} + 2 \cdot - 5} - 4 = 4$

Step 1.3

Factor $2$ out of $2 m^{2} + 2 \cdot - 5$ .

$\sqrt{2 (m^{2} - 5)} - 4 = 4$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (m^{2} - 5)}$ as ${(2 (m^{2} - 5))}^{\frac{1}{2}}$ .

${(2 (m^{2} - 5))}^{\frac{1}{2}} - 4 = 4$

Step 3

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

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Step 3.1

Add $4$ to both sides of the equation.

${(2 (m^{2} - 5))}^{\frac{1}{2}} = 4 + 4$

Step 3.2

Add $4$ and $4$ .

${(2 (m^{2} - 5))}^{\frac{1}{2}} = 8$

Step 4

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${({(2 (m^{2} - 5))}^{\frac{1}{2}})}^{2} = 8^{2}$

Step 5

Simplify the exponent.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify ${({(2 (m^{2} - 5))}^{\frac{1}{2}})}^{2}$ .

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Step 5.1.1.1

Multiply the exponents in ${({(2 (m^{2} - 5))}^{\frac{1}{2}})}^{2}$ .

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Step 5.1.1.1.1

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

${(2 (m^{2} - 5))}^{\frac{1}{2} \cdot 2} = 8^{2}$

Step 5.1.1.1.2

Cancel the common factor of $2$ .

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Step 5.1.1.1.2.1

Cancel the common factor.

${(2 (m^{2} - 5))}^{\frac{1}{2} \cdot 2} = 8^{2}$

Step 5.1.1.1.2.2

Rewrite the expression.

${(2 (m^{2} - 5))}^{1} = 8^{2}$

Step 5.1.1.2

Apply the distributive property.

${(2 m^{2} + 2 \cdot - 5)}^{1} = 8^{2}$

Step 5.1.1.3

Multiply.

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Step 5.1.1.3.1

Multiply $2$ by $- 5$ .

${(2 m^{2} - 10)}^{1} = 8^{2}$

Step 5.1.1.3.2

Simplify.

$2 m^{2} - 10 = 8^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Raise $8$ to the power of $2$ .

$2 m^{2} - 10 = 64$

Step 6

Solve for $m$ .

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Step 6.1

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

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Step 6.1.1

Add $10$ to both sides of the equation.

$2 m^{2} = 64 + 10$

Step 6.1.2

Add $64$ and $10$ .

$2 m^{2} = 74$

Step 6.2

Divide each term in $2 m^{2} = 74$ by $2$ and simplify.

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Step 6.2.1

Divide each term in $2 m^{2} = 74$ by $2$ .

$\frac{2 m^{2}}{2} = \frac{74}{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{2 m^{2}}{2} = \frac{74}{2}$

Step 6.2.2.1.2

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

$m^{2} = \frac{74}{2}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $74$ by $2$ .

$m^{2} = 37$

Step 6.3

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

$m = \pm \sqrt{37}$

Step 6.4

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

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Step 6.4.1

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

$m = \sqrt{37}$

Step 6.4.2

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

$m = - \sqrt{37}$

Step 6.4.3

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

$m = \sqrt{37}, - \sqrt{37}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$m = \sqrt{37}, - \sqrt{37}$

Decimal Form:

$m = 6.08276253 \dots, - 6.08276253 \dots$