Solve for x 2(1/49)^(x-2)=14

$2 {(\frac{1}{49})}^{x - 2} = 14$

Step 1

Divide each term in $2 {(\frac{1}{49})}^{x - 2} = 14$ by $2$ and simplify.

Divide each term in $2 {(\frac{1}{49})}^{x - 2} = 14$ by $2$ .

$\frac{2 {(\frac{1}{49})}^{x - 2}}{2} = \frac{14}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 {(\frac{1}{49})}^{x - 2}}{2} = \frac{14}{2}$

Divide ${(\frac{1}{49})}^{x - 2}$ by $1$ .

${(\frac{1}{49})}^{x - 2} = \frac{14}{2}$

Simplify the expression.

Apply the product rule to $\frac{1}{49}$ .

$\frac{1^{x - 2}}{49^{x - 2}} = \frac{14}{2}$

One to any power is one.

$\frac{1}{49^{x - 2}} = \frac{14}{2}$

Simplify the right side.

Divide $14$ by $2$ .

$\frac{1}{49^{x - 2}} = 7$

Step 2

Move $49^{x - 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$49^{- (x - 2)} = 7$

Step 3

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

$7^{2 (- (x - 2))} = 7^{1}$

Step 4

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

$2 (- (x - 2)) = 1$

Step 5

Solve for $x$ .

Simplify.

Apply the distributive property.

$2 (- x - - 2) = 1$

Multiply $- 1$ by $- 2$ .

$2 (- x + 2) = 1$

Divide each term in $2 (- x + 2) = 1$ by $- 2$ and simplify.

Divide each term in $2 (- x + 2) = 1$ by $- 2$ .

$\frac{2 (- x + 2)}{- 2} = \frac{1}{- 2}$

Simplify the left side.

Simplify the expression.

Move the negative one from the denominator of $\frac{- x + 2}{- 1}$ .

$- 1 \cdot (- x + 2) = \frac{1}{- 2}$

Rewrite $- 1 \cdot (- x + 2)$ as $- (- x + 2)$ .

$- (- x + 2) = \frac{1}{- 2}$

Apply the distributive property.

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

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Multiply $- 1$ by $2$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

Add $2$ to both sides of the equation.

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

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine $2$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $2$ .

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

Add $- 1$ and $4$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.5$

Mixed Number Form:

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

Solve for x 2(1/49)^(x-2)=14

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