Precalculus Examples

$a^{2} + {(\sqrt{19})}^{2} = {(\sqrt{24})}^{2}$

Step 1

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$a^{2} + {(19^{\frac{1}{2}})}^{2} = {\sqrt{24}}^{2}$

Step 1.2

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

$a^{2} + 19^{\frac{1}{2} \cdot 2} = {\sqrt{24}}^{2}$

Step 1.3

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

$a^{2} + 19^{\frac{2}{2}} = {\sqrt{24}}^{2}$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

$a^{2} + 19^{\frac{2}{2}} = {\sqrt{24}}^{2}$

Step 1.4.2

Rewrite the expression.

$a^{2} + 19^{1} = {\sqrt{24}}^{2}$

Step 1.5

Evaluate the exponent.

$a^{2} + 19 = {\sqrt{24}}^{2}$

Step 2

Simplify ${\sqrt{24}}^{2}$ .

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Step 2.1

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 2.1.1

Factor $4$ out of $24$ .

$a^{2} + 19 = {\sqrt{4 (6)}}^{2}$

Step 2.1.2

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

$a^{2} + 19 = {\sqrt{2^{2} \cdot 6}}^{2}$

Step 2.2

Pull terms out from under the radical.

$a^{2} + 19 = {(2 \sqrt{6})}^{2}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Apply the product rule to $2 \sqrt{6}$ .

$a^{2} + 19 = 2^{2} {\sqrt{6}}^{2}$

Step 2.3.2

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

$a^{2} + 19 = 4 {\sqrt{6}}^{2}$

Step 2.4

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$a^{2} + 19 = 4 {(6^{\frac{1}{2}})}^{2}$

Step 2.4.2

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

$a^{2} + 19 = 4 \cdot 6^{\frac{1}{2} \cdot 2}$

Step 2.4.3

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

$a^{2} + 19 = 4 \cdot 6^{\frac{2}{2}}$

Step 2.4.4

Cancel the common factor of $2$ .

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Step 2.4.4.1

Cancel the common factor.

$a^{2} + 19 = 4 \cdot 6^{\frac{2}{2}}$

Step 2.4.4.2

Rewrite the expression.

$a^{2} + 19 = 4 \cdot 6^{1}$

Step 2.4.5

Evaluate the exponent.

$a^{2} + 19 = 4 \cdot 6$

Step 2.5

Multiply $4$ by $6$ .

$a^{2} + 19 = 24$

Step 3

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

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Step 3.1

Subtract $19$ from both sides of the equation.

$a^{2} = 24 - 19$

Step 3.2

Subtract $19$ from $24$ .

$a^{2} = 5$

Step 4

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

$a = \pm \sqrt{5}$

Step 5

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

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Step 5.1

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

$a = \sqrt{5}$

Step 5.2

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

$a = - \sqrt{5}$

Step 5.3

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

$a = \sqrt{5}, - \sqrt{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$a = \sqrt{5}, - \sqrt{5}$

Decimal Form:

$a = 2.23606797 \dots, - 2.23606797 \dots$