Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

Multiply both sides of the equation by $2$ .

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

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

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Step 1.2.3.1.1.1

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

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

Step 1.2.3.1.1.2

Combine.

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

Step 1.2.3.1.1.3

Cancel the common factor of $2$ .

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Step 1.2.3.1.1.3.1

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

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

Step 1.2.3.1.1.3.2

Cancel the common factor.

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

Step 1.2.3.1.1.3.3

Rewrite the expression.

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

Step 1.2.3.1.1.4

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

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Step 1.2.3.1.1.4.1

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

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

Step 1.2.3.1.1.4.2

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

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

Step 1.2.3.1.1.5

One to any power is one.

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $2$ by $0$ .

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

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

Combine.

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

Step 2.2.2.3

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

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Step 2.2.2.3.1

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

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Step 2.2.2.3.1.1

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

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

Step 2.2.2.3.1.2

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

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

Step 2.2.2.3.2

Add $1$ and $0$ .

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

Step 2.2.2.4

Simplify $1^{1}$ .

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

Step 2.2.2.5

Simplify the denominator.

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Step 2.2.2.5.1

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

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

Step 2.2.2.5.2

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

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Step 2.2.2.5.2.1

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

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

Step 2.2.2.5.2.2

Multiply $2$ by $0$ .

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

Step 2.2.2.5.3

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

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

Step 2.2.2.5.4

Add $1$ and $0$ .

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

Step 2.2.2.6

Evaluate the exponent.

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

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{1}{2})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, \frac{1}{2})$

Step 4