Question Find the value of $x$ in the equation $\left(\frac{1}{4}\right)^{x}=2^{6 x+2}$ .

Studdy Solution

STEP 1

Assumptions
1. The equation to solve is $\left(\frac{1}{4}\right)^{x}=2^{6x+2}$ .
2. We are looking for the value of $x$ .
3. The equation involves exponential functions with a common base.

STEP 2

Recognize that $\frac{1}{4}$ is $2^{-2}$ , which allows us to rewrite the left side of the equation with a common base of 2.
$\left(\frac{1}{4}\right)^{x} = (2^{-2})^{x}$

STEP 3

Apply the power of a power rule, which states that $(a^{m})^{n} = a^{mn}$ .
$(2^{-2})^{x} = 2^{-2x}$

STEP 4

Rewrite the original equation using the common base of 2.
$2^{-2x} = 2^{6x+2}$

STEP 5

Since the bases are now the same, we can set the exponents equal to each other. This is because if $a^{m} = a^{n}$ , then $m = n$ for $a \neq 0$ .
$-2x = 6x + 2$

STEP 6

Rearrange the equation to solve for $x$ by moving all terms involving $x$ to one side of the equation.
$-2x - 6x = 2$

STEP 7

Combine like terms.
$-8x = 2$

STEP 8

Divide both sides by $-8$ to solve for $x$ .
$x = \frac{2}{-8}$

STEP 9

Simplify the fraction.
$x = -\frac{1}{4}$
The value of $x$ in the equation $\left(\frac{1}{4}\right)^{x}=2^{6x+2}$ is $x=-\frac{1}{4}$ .

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