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EXAMPLE 5 Prove that
(2^(x-1)+2^(x))/(2^(x+1)-2^(x))=(3)/(2).

EXAMPLE $5$ Prove that $\frac{2^{x-1}+2^{x}}{2^{x+1}-2^{x}}=\frac{3}{2}$ .

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Quadratic equation with complex roots

Full solution

Q. EXAMPLE

5

Prove that

\frac{2^{x-1}+2^{x}}{2^{x+1}-2^{x}}=\frac{3}{2}

.

Factor out $2^{(x-1)}$ : Factor out $2^{(x-1)}$ from the numerator and $2^x$ from the denominator. $\newline$ $\frac{2^{(x-1)} \cdot (1 + 2)}{2^x \cdot (2 - 1)}$
Simplify expression inside parentheses: Simplify the expression inside the parentheses. $(2^{(x-1)} \times 3) / (2^x \times 1)$
Divide $2^{(x-1)}$ by $2^x$ : Divide $2^{(x-1)}$ by $2^x$ in the fraction. $\newline$ $(3 \cdot 2^{(x-1)}) / 2^x = 3 \cdot (2^{(x-1)} / 2^x)$
Use property of exponents: Use the property of exponents to simplify $2^{(x-1)} / 2^x$ . $\newline$ $3 \times (2^{((x-1)-x)}) = 3 \times 2^{-1}$
Calculate $2^{-1}$ : Calculate $2^{-1}$ . $3 \times \frac{1}{2}$
Multiply by $\frac{3}{2}$ : Multiply $3$ by $\frac{1}{2}$ . $\frac{3}{2}$

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