Precalculus Examples

$\sqrt{v + 3 - 1} = 7$

Step 1

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

${\sqrt{v + 3 - 1}}^{2} = 7^{2}$

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 $\sqrt{v + 3 - 1}$ as ${(v + 3 - 1)}^{\frac{1}{2}}$ .

${({(v + 3 - 1)}^{\frac{1}{2}})}^{2} = 7^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(v + 3 - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

${(v + 3 - 1)}^{\frac{1}{2} \cdot 2} = 7^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(v + 3 - 1)}^{\frac{1}{2} \cdot 2} = 7^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(v + 3 - 1)}^{1} = 7^{2}$

Step 2.2.1.2

Subtract $1$ from $3$ .

${(v + 2)}^{1} = 7^{2}$

Step 2.2.1.3

Simplify.

$v + 2 = 7^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $7$ to the power of $2$ .

$v + 2 = 49$

Step 3

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

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Step 3.1

Subtract $2$ from both sides of the equation.

$v = 49 - 2$

Step 3.2

Subtract $2$ from $49$ .

$v = 47$