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Math  /  Algebra

Question4x⋅(1/2)3−2x=8⋅(2x)24^{x} \cdot(1 / 2)^{3-2 x}=8 \cdot\left(2^{x}\right)^{2}

Studdy Solution

STEP 1

1. The equation involves exponential expressions with different bases.
2. We will need to manipulate the equation to have the same base for comparison.
3. Simplification will involve properties of exponents.

STEP 2

1. Simplify and rewrite all terms with the same base.
2. Equate the exponents.
3. Solve for x x .

STEP 3

First, express all terms with base 2. We know: 4=22 4 = 2^2 12=2−1 \frac{1}{2} = 2^{-1} 8=23 8 = 2^3
Rewrite the equation using these expressions:
(22)x⋅(2−1)3−2x=23⋅(2x)2 (2^2)^x \cdot (2^{-1})^{3-2x} = 2^3 \cdot (2^x)^2

STEP 4

Apply the power of a power property (am)n=am⋅n(a^m)^n = a^{m \cdot n} to simplify:
22x⋅2−(3−2x)=23⋅22x 2^{2x} \cdot 2^{-(3-2x)} = 2^3 \cdot 2^{2x}
Combine the exponents on the left side:
22x−(3−2x)=23+2x 2^{2x - (3 - 2x)} = 2^{3 + 2x}
Simplify the exponent on the left:
22x−3+2x=23+2x 2^{2x - 3 + 2x} = 2^{3 + 2x} 24x−3=23+2x 2^{4x - 3} = 2^{3 + 2x}

STEP 5

Since the bases are the same, equate the exponents:
4x−3=3+2x 4x - 3 = 3 + 2x

STEP 6

Solve for x x by isolating the variable:
Subtract 2x 2x from both sides:
4x−2x−3=3 4x - 2x - 3 = 3 2x−3=3 2x - 3 = 3
Add 3 to both sides:
2x=6 2x = 6
Divide by 2:
x=3 x = 3
The value of x x is:
3 \boxed{3}

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