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Math  /  Algebra

Question7. UME 2000 Que 9
Simplify 3(2n+1)−4(2n−1)2n+1−2n\frac{3\left(2^{n+1}\right)-4\left(2^{n-1}\right)}{2^{n+1}-2^{n}} a. 2n+12^{n+1} b. 2n−12^{n-1} c. 4

Studdy Solution

STEP 1

1. We are given a fraction that needs to be simplified.
2. The expression involves powers of 2, which can be simplified using properties of exponents.
3. We assume n n is an integer.

STEP 2

1. Simplify the numerator of the fraction.
2. Simplify the denominator of the fraction.
3. Simplify the entire fraction by canceling common factors.

STEP 3

Simplify the numerator 3(2n+1)−4(2n−1) 3\left(2^{n+1}\right) - 4\left(2^{n-1}\right) .
First, express each term with the same base:
3×2n+1=3×2×2n=6×2n 3 \times 2^{n+1} = 3 \times 2 \times 2^n = 6 \times 2^n
4×2n−1=4×2n2=2×2n 4 \times 2^{n-1} = 4 \times \frac{2^n}{2} = 2 \times 2^n
Now, the numerator becomes:
6×2n−2×2n 6 \times 2^n - 2 \times 2^n
Factor out 2n 2^n :
(6−2)×2n=4×2n (6 - 2) \times 2^n = 4 \times 2^n

STEP 4

Simplify the denominator 2n+1−2n 2^{n+1} - 2^n .
Express each term with the same base:
2n+1=2×2n 2^{n+1} = 2 \times 2^n
Now, the denominator becomes:
2×2n−2n 2 \times 2^n - 2^n
Factor out 2n 2^n :
(2−1)×2n=1×2n=2n (2 - 1) \times 2^n = 1 \times 2^n = 2^n

STEP 5

Simplify the entire fraction:
4×2n2n \frac{4 \times 2^n}{2^n}
Cancel the common factor 2n 2^n :
4 4
The simplified expression is:
4 \boxed{4}

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