Math

QuestionSolve (2x)(42x)=64(2^{x})(4^{2x})=64.

Studdy Solution

STEP 1

Assumptions1. We are solving the equation for the variable xx . The equation is (x)(4x)=64\left(^{x}\right)\left(4^{ x}\right)=64

STEP 2

First, we need to simplify the equation. We can do this by expressing4 as 222^{2}.
\left(2^{x}\right)\left(2^{2}^{2 x}\right)=64

STEP 3

Now, we simplify the equation further by using the power of a power rule in exponents, which states that (am)n=amn(a^{m})^{n}=a^{m n}.
(2x)(2x)=64\left(2^{x}\right)\left(2^{ x}\right)=64

STEP 4

Next, we simplify the left side of the equation by using the product of powers rule, which states that aman=am+na^{m} \cdot a^{n}=a^{m+n}.
2x+4x=642^{x+4x}=64

STEP 5

implify the exponent on the left side of the equation.
25x=642^{5x}=64

STEP 6

Now, we need to express64 as a power of2, since we have 25x2^{5x} on the left side of the equation.
25x=262^{5x}=2^{6}

STEP 7

Since the bases on both sides of the equation are equal (both are2), we can set the exponents equal to each other.
5x=65x=6

STEP 8

Finally, we solve for xx by dividing both sides of the equation by5.
x=65x=\frac{6}{5}So, the solution to the equation (2x)(42x)=64\left(2^{x}\right)\left(4^{2 x}\right)=64 is x=65x=\frac{6}{5}.

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