Math  /  Algebra

Question2x+22x=482^{x+2}-2^{x}=48

Studdy Solution

STEP 1

1. The equation 2x+22x=48 2^{x+2} - 2^x = 48 involves exponential expressions with the same base.
2. We will use properties of exponents to simplify and solve for x x .

STEP 2

1. Simplify the equation using properties of exponents.
2. Factor out the common term.
3. Solve for x x .

STEP 3

First, use the property of exponents that states am+n=aman a^{m+n} = a^m \cdot a^n to rewrite 2x+2 2^{x+2} :
2x+2=2x22=42x 2^{x+2} = 2^x \cdot 2^2 = 4 \cdot 2^x
Substitute this back into the equation:
42x2x=48 4 \cdot 2^x - 2^x = 48

STEP 4

Factor out the common term 2x 2^x from the left side of the equation:
2x(41)=48 2^x (4 - 1) = 48
This simplifies to:
2x3=48 2^x \cdot 3 = 48

STEP 5

Divide both sides by 3 to solve for 2x 2^x :
2x=483 2^x = \frac{48}{3} 2x=16 2^x = 16
Recognize that 16 16 is a power of 2, specifically 16=24 16 = 2^4 . Therefore, we equate the exponents:
x=4 x = 4
The value of x x is:
4 \boxed{4}

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