Basic Math Examples

$2^{x} + 2^{x + 3} + 2^{x - 1} = \frac{19}{4}$

Step 1

Factor out $2^{x}$ from the expression.

$2^{x} (1 + 8 + \frac{1}{2})$

Step 2

Find the common denominator.

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Step 2.1

Write $1$ as a fraction with denominator $1$ .

$2^{x} (\frac{1}{1} + 8 + \frac{1}{2}) = \frac{19}{4}$

Step 2.2

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

$2^{x} (\frac{1}{1} \cdot \frac{2}{2} + 8 + \frac{1}{2}) = \frac{19}{4}$

Step 2.3

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

$2^{x} (\frac{2}{2} + 8 + \frac{1}{2}) = \frac{19}{4}$

Step 2.4

Write $8$ as a fraction with denominator $1$ .

$2^{x} (\frac{2}{2} + \frac{8}{1} + \frac{1}{2}) = \frac{19}{4}$

Step 2.5

Multiply $\frac{8}{1}$ by $\frac{2}{2}$ .

$2^{x} (\frac{2}{2} + \frac{8}{1} \cdot \frac{2}{2} + \frac{1}{2}) = \frac{19}{4}$

Step 2.6

Multiply $\frac{8}{1}$ by $\frac{2}{2}$ .

$2^{x} (\frac{2}{2} + \frac{8 \cdot 2}{2} + \frac{1}{2}) = \frac{19}{4}$

Step 3

Combine the numerators over the common denominator.

$2^{x} (\frac{2 + 8 \cdot 2 + 1}{2}) = \frac{19}{4}$

Step 4

Simplify the expression.

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Step 4.1

Multiply $8$ by $2$ .

$2^{x} (\frac{2 + 16 + 1}{2}) = \frac{19}{4}$

Step 4.2

Add $2$ and $16$ .

$2^{x} (\frac{18 + 1}{2}) = \frac{19}{4}$

Step 4.3

Add $18$ and $1$ .

$2^{x} (\frac{19}{2}) = \frac{19}{4}$

Step 5

Combine $2^{x}$ and $\frac{19}{2}$ .

$\frac{2^{x} \cdot 19}{2} = \frac{19}{4}$

Step 6

Cancel the common factor of $2^{x}$ and $2$ .

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Step 6.1

Factor $2$ out of $2^{x} \cdot 19$ .

$\frac{2 (2^{x - 1} \cdot 19)}{2} = \frac{19}{4}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $2$ out of $2$ .

$\frac{2 (2^{x - 1} \cdot 19)}{2 (1)} = \frac{19}{4}$

Step 6.2.2

Cancel the common factor.

$\frac{2 (2^{x - 1} \cdot 19)}{2 \cdot 1} = \frac{19}{4}$

Step 6.2.3

Rewrite the expression.

$\frac{2^{x - 1} \cdot 19}{1} = \frac{19}{4}$

Step 6.2.4

Divide $2^{x - 1} \cdot 19$ by $1$ .

$2^{x - 1} \cdot 19 = \frac{19}{4}$

Step 7

Move $19$ to the left of $2^{x - 1}$ .

$19 \cdot 2^{x - 1} = \frac{19}{4}$

Step 8

Divide each term in $19 \cdot 2^{x - 1} = \frac{19}{4}$ by $19$ and simplify.

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Step 8.1

Divide each term in $19 \cdot 2^{x - 1} = \frac{19}{4}$ by $19$ .

$\frac{19 \cdot 2^{x - 1}}{19} = \frac{\frac{19}{4}}{19}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $19$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{19 \cdot 2^{x - 1}}{19} = \frac{\frac{19}{4}}{19}$

Step 8.2.1.2

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

$2^{x - 1} = \frac{\frac{19}{4}}{19}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Multiply the numerator by the reciprocal of the denominator.

$2^{x - 1} = \frac{19}{4} \cdot \frac{1}{19}$

Step 8.3.2

Cancel the common factor of $19$ .

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Step 8.3.2.1

Cancel the common factor.

$2^{x - 1} = \frac{19}{4} \cdot \frac{1}{19}$

Step 8.3.2.2

Rewrite the expression.

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

Step 9

Raise $4$ to the power of $1$ .

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

Step 10

Move $4^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2^{x - 1} = 4^{- 1}$

Step 11

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

$2^{x - 1} = 2^{- 2}$

Step 12

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

$x - 1 = - 2$

Step 13

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

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Step 13.1

Add $1$ to both sides of the equation.

$x = - 2 + 1$

Step 13.2

Add $- 2$ and $1$ .

$x = - 1$