Basic Math Examples

${(25 k^{2})}^{\frac{1}{2}} = 15$

Step 1

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

${({(25 k^{2})}^{\frac{1}{2}})}^{2} = 15^{2}$

Step 2

Simplify the exponent.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify ${({(25 k^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.1.1.1

Multiply the exponents in ${({(25 k^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.1.1.1.1

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

${(25 k^{2})}^{\frac{1}{2} \cdot 2} = 15^{2}$

Step 2.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.1.1.2.1

Cancel the common factor.

${(25 k^{2})}^{\frac{1}{2} \cdot 2} = 15^{2}$

Step 2.1.1.1.2.2

Rewrite the expression.

${(25 k^{2})}^{1} = 15^{2}$

Step 2.1.1.2

Simplify.

$25 k^{2} = 15^{2}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Raise $15$ to the power of $2$ .

$25 k^{2} = 225$

Step 3

Solve for $k$ .

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Step 3.1

Divide each term in $25 k^{2} = 225$ by $25$ and simplify.

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Step 3.1.1

Divide each term in $25 k^{2} = 225$ by $25$ .

$\frac{25 k^{2}}{25} = \frac{225}{25}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $25$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{25 k^{2}}{25} = \frac{225}{25}$

Step 3.1.2.1.2

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

$k^{2} = \frac{225}{25}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $225$ by $25$ .

$k^{2} = 9$

Step 3.2

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

$k = \pm \sqrt{9}$

Step 3.3

Simplify $\pm \sqrt{9}$ .

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Step 3.3.1

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

$k = \pm \sqrt{3^{2}}$

Step 3.3.2

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

$k = \pm 3$

Step 3.4

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

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Step 3.4.1

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

$k = 3$

Step 3.4.2

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

$k = - 3$

Step 3.4.3

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

$k = 3, - 3$