Simplify the following expression: (2^(x+1) + 2^(x+2)) / (2^x + 2^(x+2)) + (2^(-1) + 3^(-1))^2 + [(2.2^n + 6.2^(n-1)) / 5.4^n]^(-1) + sqrt(5^2 - 4^2) + (5.3^n - 9.3^(n-2))
Understand the Problem
The question requires simplifying a set of mathematical expressions involving exponents, fractions, and square roots. Each part needs to be evaluated and simplified step by step.
Answer
$$ \begin{aligned} \frac{2^{x+1} +2^{x+2}}{2^{x}+2^{x+2}} &= \frac{6}{5} \\ (2^{-1} +3^{-1})^2 &= \frac{25}{36} \\ [\frac{2 \cdot 2^{n} + 6 \cdot 2^{n-1}}{5 \cdot 4^{n}}]^{-1} &= 2^n \\ \sqrt{5^{2}-4^{2}} + (5 \cdot 3^{n} - 9 \cdot 3^{n-2}) &= 3 + 4 \cdot 3^n \end{aligned} $$
Answer for screen readers
$$ \begin{aligned} \frac{2^{x+1} +2^{x+2}}{2^{x}+2^{x+2}} &= \frac{6}{5} \ (2^{-1} +3^{-1})^2 &= \frac{25}{36} \ [\frac{2 \cdot 2^{n} + 6 \cdot 2^{n-1}}{5 \cdot 4^{n}}]^{-1} &= 2^n \ \sqrt{5^{2}-4^{2}} + (5 \cdot 3^{n} - 9 \cdot 3^{n-2}) &= 3 + 4 \cdot 3^n \end{aligned} $$
Steps to Solve
- Simplify $\frac{2^{x+1} +2^{x+2}}{2^{x}+2^{x+2}}$
Factor out common terms: $$ \frac{2^{x+1} +2^{x+2}}{2^{x}+2^{x+2}} = \frac{2^x \cdot 2^1 + 2^x \cdot 2^2}{2^x + 2^x \cdot 2^2} = \frac{2^x(2 + 4)}{2^x(1 + 4)} = \frac{6}{5} $$
- Simplify $(2^{-1} +3^{-1})^2$
Rewrite negative exponents as fractions, and simplify inside the parenthesis: $$ (2^{-1} +3^{-1})^2 = (\frac{1}{2} + \frac{1}{3})^2 = (\frac{3}{6} + \frac{2}{6})^2 = (\frac{5}{6})^2 = \frac{25}{36} $$
- Simplify $[\frac{2 \cdot 2^{n} + 6 \cdot 2^{n-1}}{5 \cdot 4^{n}}]^{-1}$
Rewrite $2^{n-1}$ as $\frac{2^n}{2}$ and $4^n$ as $(2^2)^n = 2^{2n}$, then simplify: $$ [\frac{2 \cdot 2^{n} + 6 \cdot 2^{n-1}}{5 \cdot 4^{n}}]^{-1} = [\frac{2 \cdot 2^{n} + 6 \cdot \frac{2^{n}}{2}}{5 \cdot 2^{2n}}]^{-1} = [\frac{2 \cdot 2^{n} + 3 \cdot 2^{n}}{5 \cdot 2^{2n}}]^{-1} = [\frac{5 \cdot 2^{n}}{5 \cdot 2^{2n}}]^{-1} = [\frac{1}{2^{n}}]^{-1}= 2^n $$
- Simplify $\sqrt{5^{2}-4^{2}} + (5 \cdot 3^{n} - 9 \cdot 3^{n-2})$
Simplify the square root and rewrite $3^{n-2}$ as $\frac{3^n}{3^2} = \frac{3^n}{9}$: $$ \sqrt{5^{2}-4^{2}} + (5 \cdot 3^{n} - 9 \cdot 3^{n-2}) = \sqrt{25-16} + (5 \cdot 3^{n} - 9 \cdot \frac{3^{n}}{9}) = \sqrt{9} + (5 \cdot 3^{n} - 3^{n}) = 3 + 4 \cdot 3^n $$
$$ \begin{aligned} \frac{2^{x+1} +2^{x+2}}{2^{x}+2^{x+2}} &= \frac{6}{5} \ (2^{-1} +3^{-1})^2 &= \frac{25}{36} \ [\frac{2 \cdot 2^{n} + 6 \cdot 2^{n-1}}{5 \cdot 4^{n}}]^{-1} &= 2^n \ \sqrt{5^{2}-4^{2}} + (5 \cdot 3^{n} - 9 \cdot 3^{n-2}) &= 3 + 4 \cdot 3^n \end{aligned} $$
More Information
Each expression was simplified using exponent rules, fraction arithmetic, and basic algebraic manipulation.
Tips
A common mistake would be incorrectly applying exponent rules, especially when dealing with negative exponents or fractional exponents. Another mistake would be incorrectly factoring out common terms. Care should be taken in each step to ensure the rules are correctly applied.
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