Precalculus Examples

$y - 2 = \sqrt{t}$

Step 1

Rewrite the equation as $\sqrt{t} = y - 2$ .

$\sqrt{t} = y - 2$

Step 2

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

${\sqrt{t}}^{2} = {(y - 2)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

${(t^{\frac{1}{2}})}^{2} = {(y - 2)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

$t^{\frac{1}{2} \cdot 2} = {(y - 2)}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$t^{\frac{1}{2} \cdot 2} = {(y - 2)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

$t^{1} = {(y - 2)}^{2}$

Step 3.2.1.2

Simplify.

$t = {(y - 2)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(y - 2)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(y - 2)}^{2}$ as $(y - 2) (y - 2)$ .

$t = (y - 2) (y - 2)$

Step 3.3.1.2

Expand $(y - 2) (y - 2)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$t = y (y - 2) - 2 (y - 2)$

Step 3.3.1.2.2

Apply the distributive property.

$t = y \cdot y + y \cdot - 2 - 2 (y - 2)$

Step 3.3.1.2.3

Apply the distributive property.

$t = y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $y$ by $y$ .

$t = y^{2} + y \cdot - 2 - 2 y - 2 \cdot - 2$

Step 3.3.1.3.1.2

Move $- 2$ to the left of $y$ .

$t = y^{2} - 2 \cdot y - 2 y - 2 \cdot - 2$

Step 3.3.1.3.1.3

Multiply $- 2$ by $- 2$ .

$t = y^{2} - 2 y - 2 y + 4$

Step 3.3.1.3.2

Subtract $2 y$ from $- 2 y$ .

$t = y^{2} - 4 y + 4$