Algebra Examples

$80 = 1.25 {(\frac{1}{2})}^{- 2 x - 5}$

Step 1

Rewrite the equation as $1.25 {(\frac{1}{2})}^{- 2 x - 5} = 80$ .

$1.25 {(\frac{1}{2})}^{- 2 x - 5} = 80$

Step 2

Divide each term in $1.25 {(\frac{1}{2})}^{- 2 x - 5} = 80$ by $1.25$ and simplify.

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Step 2.1

Divide each term in $1.25 {(\frac{1}{2})}^{- 2 x - 5} = 80$ by $1.25$ .

$\frac{1.25 {(\frac{1}{2})}^{- 2 x - 5}}{1.25} = \frac{80}{1.25}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $1.25$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{1.25 {(\frac{1}{2})}^{- 2 x - 5}}{1.25} = \frac{80}{1.25}$

Step 2.2.1.2

Divide ${(\frac{1}{2})}^{- 2 x - 5}$ by $1$ .

${(\frac{1}{2})}^{- 2 x - 5} = \frac{80}{1.25}$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Apply the product rule to $\frac{1}{2}$ .

$\frac{1^{- 2 x - 5}}{2^{- 2 x - 5}} = \frac{80}{1.25}$

Step 2.2.2.2

One to any power is one.

$\frac{1}{2^{- 2 x - 5}} = \frac{80}{1.25}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $80$ by $1.25$ .

$\frac{1}{2^{- 2 x - 5}} = 64$

Step 3

Move $2^{- 2 x - 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$2^{- (- 2 x - 5)} = 64$

Step 4

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

$2^{- (- 2 x - 5)} = 2^{6}$

Step 5

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

$- (- 2 x - 5) = 6$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $- (- 2 x - 5)$ .

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Step 6.1.1

Apply the distributive property.

$- (- 2 x) - - 5 = 6$

Step 6.1.2

Multiply.

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Step 6.1.2.1

Multiply $- 2$ by $- 1$ .

$2 x - - 5 = 6$

Step 6.1.2.2

Multiply $- 1$ by $- 5$ .

$2 x + 5 = 6$

Step 6.2

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

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Step 6.2.1

Subtract $5$ from both sides of the equation.

$2 x = 6 - 5$

Step 6.2.2

Subtract $5$ from $6$ .

$2 x = 1$

Step 6.3

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 6.3.1

Divide each term in $2 x = 1$ by $2$ .

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

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $2$ .

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Step 6.3.2.1.1

Cancel the common factor.

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

Step 6.3.2.1.2

Divide $x$ by $1$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.5$