Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

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

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

Step 1.2.2.2

One to any power is one.

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

Step 1.2.2.3

Combine $8$ and $\frac{1}{4^{x}}$ .

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

Step 1.2.3

Add $1$ to both sides of the equation.

$\frac{8}{4^{x}} = 1$

Step 1.2.4

Multiply both sides by $4^{x}$ .

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

Step 1.2.5

Simplify.

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Step 1.2.5.1

Simplify the left side.

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Step 1.2.5.1.1

Cancel the common factor of $4^{x}$ .

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Step 1.2.5.1.1.1

Cancel the common factor.

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

Step 1.2.5.1.1.2

Rewrite the expression.

$8 = 1 \cdot 4^{x}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Multiply $4^{x}$ by $1$ .

$8 = 4^{x}$

Step 1.2.6

Solve for $x$ .

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Step 1.2.6.1

Rewrite the equation as $4^{x} = 8$ .

$4^{x} = 8$

Step 1.2.6.2

Create equivalent expressions in the equation that all have equal bases.

$2^{2 x} = 2^{3}$

Step 1.2.6.3

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$2 x = 3$

Step 1.2.6.4

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 1.2.6.4.1

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Step 1.2.6.4.2

Simplify the left side.

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Step 1.2.6.4.2.1

Cancel the common factor of $2$ .

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Step 1.2.6.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Step 1.2.6.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{3}{2}, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

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

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

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

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

Step 2.2.2.1.2

Anything raised to $0$ is $1$ .

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

Step 2.2.2.1.3

Anything raised to $0$ is $1$ .

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

Step 2.2.2.1.4

Cancel the common factor of $1$ .

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Step 2.2.2.1.4.1

Cancel the common factor.

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

Step 2.2.2.1.4.2

Rewrite the expression.

$y = 8 \cdot 1 - 1$

Step 2.2.2.1.5

Multiply $8$ by $1$ .

$y = 8 - 1$

Step 2.2.2.2

Subtract $1$ from $8$ .

$y = 7$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 7)$

Step 3

List the intersections.

x-intercept(s): $(\frac{3}{2}, 0)$

y-intercept(s): $(0, 7)$

Step 4