Algebra Examples

$y = \frac{1}{2} \cdot {(\frac{1}{4})}^{x}$

Step 1

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

$y = \frac{1}{2} \cdot {(\frac{1}{4})}^{- 2}$

Step 2

Solve the equation for $y$ .

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Step 2.1

Remove parentheses.

$y = \frac{1}{2} \cdot {(\frac{1}{4})}^{- 2}$

Step 2.2

Simplify $\frac{1}{2} \cdot {(\frac{1}{4})}^{- 2}$ .

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Step 2.2.1

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

$y = \frac{1}{2} \cdot 4^{2}$

Step 2.2.2

Raise $4$ to the power of $2$ .

$y = \frac{1}{2} \cdot 16$

Step 2.2.3

Cancel the common factor of $2$ .

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Step 2.2.3.1

Factor $2$ out of $16$ .

$y = \frac{1}{2} \cdot (2 (8))$

Step 2.2.3.2

Cancel the common factor.

$y = \frac{1}{2} \cdot (2 \cdot 8)$

Step 2.2.3.3

Rewrite the expression.

$y = 8$

Step 3

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

$y = \frac{1}{2} \cdot {(\frac{1}{4})}^{- 1}$

Step 4

Solve the equation for $y$ .

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Step 4.1

Remove parentheses.

$y = \frac{1}{2} \cdot {(\frac{1}{4})}^{- 1}$

Step 4.2

Simplify $\frac{1}{2} \cdot {(\frac{1}{4})}^{- 1}$ .

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Step 4.2.1

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

$y = \frac{1}{2} \cdot 4$

Step 4.2.2

Cancel the common factor of $2$ .

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Step 4.2.2.1

Factor $2$ out of $4$ .

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

$y = 2$

Step 5

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

$y = \frac{1}{2} \cdot {(\frac{1}{4})}^{0}$

Step 6

Solve the equation for $y$ .

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Step 6.1

Remove parentheses.

$y = \frac{1}{2} \cdot {(\frac{1}{4})}^{0}$

Step 6.2

Simplify $\frac{1}{2} \cdot {(\frac{1}{4})}^{0}$ .

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Step 6.2.1

Apply the product rule to $\frac{1}{4}$ .

$y = \frac{1}{2} \cdot \frac{1^{0}}{4^{0}}$

Step 6.2.2

Combine.

$y = \frac{1 \cdot 1^{0}}{2 \cdot 4^{0}}$

Step 6.2.3

Multiply $1$ by $1^{0}$ by adding the exponents.

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Step 6.2.3.1

Multiply $1$ by $1^{0}$ .

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Step 6.2.3.1.1

Raise $1$ to the power of $1$ .

$y = \frac{1^{1} \cdot 1^{0}}{2 \cdot 4^{0}}$

Step 6.2.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1^{1 + 0}}{2 \cdot 4^{0}}$

Step 6.2.3.2

Add $1$ and $0$ .

$y = \frac{1^{1}}{2 \cdot 4^{0}}$

Step 6.2.4

Simplify $1^{1}$ .

$y = \frac{1}{2 \cdot 4^{0}}$

Step 6.2.5

Simplify the denominator.

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Step 6.2.5.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{1}{2 \cdot {(2^{2})}^{0}}$

Step 6.2.5.2

Multiply the exponents in ${(2^{2})}^{0}$ .

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Step 6.2.5.2.1

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

$y = \frac{1}{2 \cdot 2^{2 \cdot 0}}$

Step 6.2.5.2.2

Multiply $2$ by $0$ .

$y = \frac{1}{2 \cdot 2^{0}}$

Step 6.2.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.2.5.4

Add $1$ and $0$ .

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

Step 6.2.6

Evaluate the exponent.

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

Step 7

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

$y = \frac{1}{2} \cdot {(\frac{1}{4})}^{1}$

Step 8

Solve the equation for $y$ .

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Step 8.1

Remove parentheses.

$y = \frac{1}{2} \cdot (\frac{1}{4})$

Step 8.2

Multiply $\frac{1}{2} (\frac{1}{4})$ .

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Step 8.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

$y = \frac{1}{2 \cdot 4}$

Step 8.2.2

Multiply $2$ by $4$ .

$y = \frac{1}{8}$

Step 9

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

$y = \frac{1}{2} \cdot {(\frac{1}{4})}^{2}$

Step 10

Solve the equation for $y$ .

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Step 10.1

Remove parentheses.

$y = \frac{1}{2} \cdot {(\frac{1}{4})}^{2}$

Step 10.2

Simplify $\frac{1}{2} \cdot {(\frac{1}{4})}^{2}$ .

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Step 10.2.1

Apply the product rule to $\frac{1}{4}$ .

$y = \frac{1}{2} \cdot \frac{1^{2}}{4^{2}}$

Step 10.2.2

Combine.

$y = \frac{1 \cdot 1^{2}}{2 \cdot 4^{2}}$

Step 10.2.3

Multiply $1$ by $1^{2}$ by adding the exponents.

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Step 10.2.3.1

Multiply $1$ by $1^{2}$ .

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Step 10.2.3.1.1

Raise $1$ to the power of $1$ .

$y = \frac{1^{1} \cdot 1^{2}}{2 \cdot 4^{2}}$

Step 10.2.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1^{1 + 2}}{2 \cdot 4^{2}}$

Step 10.2.3.2

Add $1$ and $2$ .

$y = \frac{1^{3}}{2 \cdot 4^{2}}$

Step 10.2.4

One to any power is one.

$y = \frac{1}{2 \cdot 4^{2}}$

Step 10.2.5

Simplify the denominator.

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Step 10.2.5.1

Rewrite $4$ as $2^{2}$ .

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

Step 10.2.5.2

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

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Step 10.2.5.2.1

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

$y = \frac{1}{2 \cdot 2^{2 \cdot 2}}$

Step 10.2.5.2.2

Multiply $2$ by $2$ .

$y = \frac{1}{2 \cdot 2^{4}}$

Step 10.2.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1}{2^{1 + 4}}$

Step 10.2.5.4

Add $1$ and $4$ .

$y = \frac{1}{2^{5}}$

Step 10.2.6

Raise $2$ to the power of $5$ .

$y = \frac{1}{32}$

Step 11

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

$\begin{array}{c} x & y \\ - 2 & 8 \\ - 1 & 2 \\ 0 & \frac{1}{2} \\ 1 & \frac{1}{8} \\ 2 & \frac{1}{32} \end{array}$

Step 12