Trigonometry Examples

$5 {(\frac{1}{4})}^{x} - 20 = 20$

Step 1

Simplify each term.

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Step 1.1

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

$5 \frac{1^{x}}{4^{x}} - 20 = 20$

Step 1.2

One to any power is one.

$5 \frac{1}{4^{x}} - 20 = 20$

Step 1.3

Combine $5$ and $\frac{1}{4^{x}}$ .

$\frac{5}{4^{x}} - 20 = 20$

Step 2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.1

Add $20$ to both sides of the equation.

$\frac{5}{4^{x}} = 20 + 20$

Step 2.2

Add $20$ and $20$ .

$\frac{5}{4^{x}} = 40$

Step 3

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

$\frac{5}{4^{x}} \cdot 4^{x} = 40 \cdot 4^{x}$

Step 4

Simplify the left side.

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Step 4.1

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

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Step 4.1.1

Cancel the common factor.

$\frac{5}{4^{x}} \cdot 4^{x} = 40 \cdot 4^{x}$

Step 4.1.2

Rewrite the expression.

$5 = 40 \cdot 4^{x}$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $40 \cdot 4^{x} = 5$ .

$40 \cdot 4^{x} = 5$

Step 5.2

Divide each term in $40 \cdot 4^{x} = 5$ by $40$ and simplify.

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Step 5.2.1

Divide each term in $40 \cdot 4^{x} = 5$ by $40$ .

$\frac{40 \cdot 4^{x}}{40} = \frac{5}{40}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $40$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{40 \cdot 4^{x}}{40} = \frac{5}{40}$

Step 5.2.2.1.2

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

$4^{x} = \frac{5}{40}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Cancel the common factor of $5$ and $40$ .

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Step 5.2.3.1.1

Factor $5$ out of $5$ .

$4^{x} = \frac{5 (1)}{40}$

Step 5.2.3.1.2

Cancel the common factors.

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Step 5.2.3.1.2.1

Factor $5$ out of $40$ .

$4^{x} = \frac{5 \cdot 1}{5 \cdot 8}$

Step 5.2.3.1.2.2

Cancel the common factor.

$4^{x} = \frac{5 \cdot 1}{5 \cdot 8}$

Step 5.2.3.1.2.3

Rewrite the expression.

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

Step 5.3

Raise $8$ to the power of $1$ .

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

Step 5.4

Move $8^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$4^{x} = 8^{- 1}$

Step 5.5

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

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

Step 5.6

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

$2 x = - 3$

Step 5.7

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

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Step 5.7.1

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

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

Step 5.7.2

Simplify the left side.

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Step 5.7.2.1

Cancel the common factor of $2$ .

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Step 5.7.2.1.1

Cancel the common factor.

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

Step 5.7.2.1.2

Divide $x$ by $1$ .

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

Step 5.7.3

Simplify the right side.

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Step 5.7.3.1

Move the negative in front of the fraction.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 1.5$

Mixed Number Form:

$x = - 1 \frac{1}{2}$