Trigonometry Examples

$2^{x - 3} \cdot 2^{- 2 x} = 16$

Step 1

Multiply $2^{x - 3}$ by $2^{- 2 x}$ by adding the exponents.

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Step 1.1

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

$2^{x - 3 - 2 x} = 16$

Step 1.2

Subtract $2 x$ from $x$ .

$2^{- x - 3} = 16$

Step 2

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

$2^{- x - 3} = 2^{4}$

Step 3

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

$- x - 3 = 4$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Add $3$ to both sides of the equation.

$- x = 4 + 3$

Step 4.1.2

Add $4$ and $3$ .

$- x = 7$

Step 4.2

Divide each term in $- x = 7$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- x = 7$ by $- 1$ .

$\frac{- x}{- 1} = \frac{7}{- 1}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{7}{- 1}$

Step 4.2.2.2

Divide $x$ by $1$ .

$x = \frac{7}{- 1}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $7$ by $- 1$ .

$x = - 7$