Algebra Examples

$y = {(\frac{2}{3})}^{x} + 2$

Step 1

Substitute $- 2$ for $x$ and find the result for $y$ .

$y = {(\frac{2}{3})}^{- 2} + 2$

Step 2

Solve the equation for $y$ .

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Step 2.1

Remove parentheses.

$y = {(\frac{2}{3})}^{- 2} + 2$

Step 2.2

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

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Change the sign of the exponent by rewriting the base as its reciprocal.

$y = {(\frac{3}{2})}^{2} + 2$

Step 2.2.1.2

Apply the product rule to $\frac{3}{2}$ .

$y = \frac{3^{2}}{2^{2}} + 2$

Step 2.2.1.3

Raise $3$ to the power of $2$ .

$y = \frac{9}{2^{2}} + 2$

Step 2.2.1.4

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

$y = \frac{9}{4} + 2$

Step 2.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{9}{4} + 2 \cdot \frac{4}{4}$

Step 2.2.3

Combine $2$ and $\frac{4}{4}$ .

$y = \frac{9}{4} + \frac{2 \cdot 4}{4}$

Step 2.2.4

Combine the numerators over the common denominator.

$y = \frac{9 + 2 \cdot 4}{4}$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $2$ by $4$ .

$y = \frac{9 + 8}{4}$

Step 2.2.5.2

Add $9$ and $8$ .

$y = \frac{17}{4}$

Step 3

Substitute $- 1$ for $x$ and find the result for $y$ .

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

Step 4

Solve the equation for $y$ .

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Step 4.1

Remove parentheses.

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

Step 4.2

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

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Step 4.2.1

Change the sign of the exponent by rewriting the base as its reciprocal.

$y = \frac{3}{2} + 2$

Step 4.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{3}{2} + 2 \cdot \frac{2}{2}$

Step 4.2.3

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

$y = \frac{3}{2} + \frac{2 \cdot 2}{2}$

Step 4.2.4

Combine the numerators over the common denominator.

$y = \frac{3 + 2 \cdot 2}{2}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $2$ by $2$ .

$y = \frac{3 + 4}{2}$

Step 4.2.5.2

Add $3$ and $4$ .

$y = \frac{7}{2}$

Step 5

Substitute $0$ for $x$ and find the result for $y$ .

$y = {(\frac{2}{3})}^{0} + 2$

Step 6

Solve the equation for $y$ .

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Step 6.1

Remove parentheses.

$y = {(\frac{2}{3})}^{0} + 2$

Step 6.2

Simplify ${(\frac{2}{3})}^{0} + 2$ .

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Apply the product rule to $\frac{2}{3}$ .

$y = \frac{2^{0}}{3^{0}} + 2$

Step 6.2.1.2

Anything raised to $0$ is $1$ .

$y = \frac{1}{3^{0}} + 2$

Step 6.2.1.3

Anything raised to $0$ is $1$ .

$y = \frac{1}{1} + 2$

Step 6.2.1.4

Divide $1$ by $1$ .

$y = 1 + 2$

Step 6.2.2

Add $1$ and $2$ .

$y = 3$

Step 7

Substitute $1$ for $x$ and find the result for $y$ .

$y = {(\frac{2}{3})}^{1} + 2$

Step 8

Solve the equation for $y$ .

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Step 8.1

Remove parentheses.

$y = (\frac{2}{3}) + 2$

Step 8.2

Simplify $(\frac{2}{3}) + 2$ .

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Step 8.2.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{2}{3} + 2 \cdot \frac{3}{3}$

Step 8.2.2

Combine $2$ and $\frac{3}{3}$ .

$y = \frac{2}{3} + \frac{2 \cdot 3}{3}$

Step 8.2.3

Combine the numerators over the common denominator.

$y = \frac{2 + 2 \cdot 3}{3}$

Step 8.2.4

Simplify the numerator.

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Step 8.2.4.1

Multiply $2$ by $3$ .

$y = \frac{2 + 6}{3}$

Step 8.2.4.2

Add $2$ and $6$ .

$y = \frac{8}{3}$

Step 9

Substitute $2$ for $x$ and find the result for $y$ .

$y = {(\frac{2}{3})}^{2} + 2$

Step 10

Solve the equation for $y$ .

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Step 10.1

Remove parentheses.

$y = {(\frac{2}{3})}^{2} + 2$

Step 10.2

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

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Step 10.2.1

Simplify each term.

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Step 10.2.1.1

Apply the product rule to $\frac{2}{3}$ .

$y = \frac{2^{2}}{3^{2}} + 2$

Step 10.2.1.2

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

$y = \frac{4}{3^{2}} + 2$

Step 10.2.1.3

Raise $3$ to the power of $2$ .

$y = \frac{4}{9} + 2$

Step 10.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = \frac{4}{9} + 2 \cdot \frac{9}{9}$

Step 10.2.3

Combine $2$ and $\frac{9}{9}$ .

$y = \frac{4}{9} + \frac{2 \cdot 9}{9}$

Step 10.2.4

Combine the numerators over the common denominator.

$y = \frac{4 + 2 \cdot 9}{9}$

Step 10.2.5

Simplify the numerator.

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Step 10.2.5.1

Multiply $2$ by $9$ .

$y = \frac{4 + 18}{9}$

Step 10.2.5.2

Add $4$ and $18$ .

$y = \frac{22}{9}$

Step 11

This is a table of possible values to use when graphing the equation.

$\begin{array}{c} x & y \\ - 2 & \frac{17}{4} \\ - 1 & \frac{7}{2} \\ 0 & 3 \\ 1 & \frac{8}{3} \\ 2 & \frac{22}{9} \end{array}$

Step 12