Solve for a 17^2 = square root of 8^2+a^2

$17^{2} = \sqrt{8^{2} + a^{2}}$

Step 1

Rewrite the equation as $\sqrt{8^{2} + a^{2}} = 17^{2}$ .

$\sqrt{8^{2} + a^{2}} = 17^{2}$

Step 2

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

${\sqrt{8^{2} + a^{2}}}^{2} = {(17^{2})}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8^{2} + a^{2}}$ as ${(8^{2} + a^{2})}^{\frac{1}{2}}$ .

${({(8^{2} + a^{2})}^{\frac{1}{2}})}^{2} = {(17^{2})}^{2}$

Simplify the left side.

Simplify ${({(8^{2} + a^{2})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(8^{2} + a^{2})}^{\frac{1}{2}})}^{2}$ .

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

${(8^{2} + a^{2})}^{\frac{1}{2} \cdot 2} = {(17^{2})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(8^{2} + a^{2})}^{\frac{1}{2} \cdot 2} = {(17^{2})}^{2}$

Rewrite the expression.

${(8^{2} + a^{2})}^{1} = {(17^{2})}^{2}$

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

${(64 + a^{2})}^{1} = {(17^{2})}^{2}$

Simplify.

$64 + a^{2} = {(17^{2})}^{2}$

Simplify the right side.

Simplify ${(17^{2})}^{2}$ .

Multiply the exponents in ${(17^{2})}^{2}$ .

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

$64 + a^{2} = 17^{2 \cdot 2}$

Multiply $2$ by $2$ .

$64 + a^{2} = 17^{4}$

Raise $17$ to the power of $4$ .

$64 + a^{2} = 83521$

Step 4

Solve for $a$ .

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

Subtract $64$ from both sides of the equation.

$a^{2} = 83521 - 64$

Subtract $64$ from $83521$ .

$a^{2} = 83457$

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

$a = \pm \sqrt{83457}$

Simplify $\pm \sqrt{83457}$ .

Rewrite $83457$ as $3^{2} \cdot 9273$ .

Factor $9$ out of $83457$ .

$a = \pm \sqrt{9 (9273)}$

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

$a = \pm \sqrt{3^{2} \cdot 9273}$

Pull terms out from under the radical.

$a = \pm 3 \sqrt{9273}$

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

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

$a = 3 \sqrt{9273}$

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

$a = - 3 \sqrt{9273}$

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

$a = 3 \sqrt{9273}, - 3 \sqrt{9273}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$a = 3 \sqrt{9273}, - 3 \sqrt{9273}$

Decimal Form:

$a = 288.88925213 \dots, - 288.88925213 \dots$

Solve for a 17^2 = square root of 8^2+a^2

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