Algebra Examples

$f (x) = \frac{1}{8} {(\frac{1}{4})}^{x - 2}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{1}{8} \cdot {(\frac{1}{4})}^{x - 2}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{1}{8} \cdot {(\frac{1}{4})}^{x - 2} = 0$ .

$\frac{1}{8} \cdot {(\frac{1}{4})}^{x - 2} = 0$

Step 1.2.2

Multiply both sides of the equation by $8$ .

$8 (\frac{1}{8} \cdot {(\frac{1}{4})}^{x - 2}) = 8 \cdot 0$

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Simplify $8 (\frac{1}{8} \cdot {(\frac{1}{4})}^{x - 2})$ .

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Step 1.2.3.1.1.1

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

$8 (\frac{1}{8} \cdot \frac{1^{x - 2}}{4^{x - 2}}) = 8 \cdot 0$

Step 1.2.3.1.1.2

Combine.

$8 \frac{1 \cdot 1^{x - 2}}{8 \cdot 4^{x - 2}} = 8 \cdot 0$

Step 1.2.3.1.1.3

Cancel the common factor of $8$ .

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Step 1.2.3.1.1.3.1

Factor $8$ out of $8 \cdot 4^{x - 2}$ .

$8 \frac{1 \cdot 1^{x - 2}}{8 (4^{x - 2})} = 8 \cdot 0$

Step 1.2.3.1.1.3.2

Cancel the common factor.

$8 \frac{1 \cdot 1^{x - 2}}{8 \cdot 4^{x - 2}} = 8 \cdot 0$

Step 1.2.3.1.1.3.3

Rewrite the expression.

$\frac{1 \cdot 1^{x - 2}}{4^{x - 2}} = 8 \cdot 0$

Step 1.2.3.1.1.4

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

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Step 1.2.3.1.1.4.1

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

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Step 1.2.3.1.1.4.1.1

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

$\frac{1^{1} \cdot 1^{x - 2}}{4^{x - 2}} = 8 \cdot 0$

Step 1.2.3.1.1.4.1.2

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

$\frac{1^{1 + x - 2}}{4^{x - 2}} = 8 \cdot 0$

Step 1.2.3.1.1.4.2

Subtract $2$ from $1$ .

$\frac{1^{x - 1}}{4^{x - 2}} = 8 \cdot 0$

Step 1.2.3.1.1.5

One to any power is one.

$\frac{1}{4^{x - 2}} = 8 \cdot 0$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $8$ by $0$ .

$\frac{1}{4^{x - 2}} = 0$

Step 1.2.4

Set the numerator equal to zero.

$1 = 0$

Step 1.2.5

Since $1 \neq 0$ , there are no solutions.

No solution

Step 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{1}{8} \cdot {(\frac{1}{4})}^{(0) - 2}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{1}{8} \cdot {(\frac{1}{4})}^{0 - 2}$

Step 2.2.2

Remove parentheses.

$y = \frac{1}{8} \cdot {(\frac{1}{4})}^{(0) - 2}$

Step 2.2.3

Simplify $\frac{1}{8} \cdot {(\frac{1}{4})}^{(0) - 2}$ .

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Step 2.2.3.1

Subtract $2$ from $0$ .

$y = \frac{1}{8} \cdot {(\frac{1}{4})}^{- 2}$

Step 2.2.3.2

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

$y = \frac{1}{8} \cdot 4^{2}$

Step 2.2.3.3

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

$y = \frac{1}{8} \cdot 16$

Step 2.2.3.4

Cancel the common factor of $8$ .

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Step 2.2.3.4.1

Factor $8$ out of $16$ .

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

Step 2.2.3.4.2

Cancel the common factor.

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

Step 2.2.3.4.3

Rewrite the expression.

$y = 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 2)$

Step 4