Math Problem Statement

(16 * 2 ^ (n + 1) - 4 * 2 ^ n)/(16 * 2 ^ (n + 1) - 2 * 2 ^ (n + 2)) =

(a) - 1/4

(b) - 1/2

1/2 (0)

(P) 1/4

Solution

To simplify the given expression:

162n+142n162n+122n+2\frac{16 \cdot 2^{n+1} - 4 \cdot 2^n}{16 \cdot 2^{n+1} - 2 \cdot 2^{n+2}}

we can start by simplifying both the numerator and the denominator.

Step 1: Simplify the numerator

The numerator is:

162n+142n16 \cdot 2^{n+1} - 4 \cdot 2^n

Factor out the common term 2n2^n:

=2n(1624)= 2^n (16 \cdot 2 - 4)

Calculate the terms inside the parentheses:

162=32,324=2816 \cdot 2 = 32, \quad 32 - 4 = 28

So, the simplified numerator is:

2n28=282n2^n \cdot 28 = 28 \cdot 2^n

Step 2: Simplify the denominator

The denominator is:

162n+122n+216 \cdot 2^{n+1} - 2 \cdot 2^{n+2}

Rewrite 2n+22^{n+2} as 42n4 \cdot 2^n and factor out the common term 2n2^n:

=2n(16224)= 2^n (16 \cdot 2 - 2 \cdot 4)

Calculate the terms inside the parentheses:

162=32,24=8,328=2416 \cdot 2 = 32, \quad 2 \cdot 4 = 8, \quad 32 - 8 = 24

So, the simplified denominator is:

2n24=242n2^n \cdot 24 = 24 \cdot 2^n

Step 3: Simplify the fraction

Now the expression is:

282n242n\frac{28 \cdot 2^n}{24 \cdot 2^n}

Since 2n2^n appears in both the numerator and the denominator, we can cancel it:

2824\frac{28}{24}

Simplify the fraction by dividing both the numerator and the denominator by 4:

28÷424÷4=76\frac{28 \div 4}{24 \div 4} = \frac{7}{6}

Conclusion

The simplified value of the expression is 76\frac{7}{6}, which does not match any of the provided options exactly. This indicates a possible error in the options given or a misunderstanding of the problem context.

If the answer choices or context were incorrect, here’s the correct simplified form:

76\frac{7}{6}

Do you have any more details or need further clarifications?

Relative Questions:

  1. What if the original expression had different coefficients?
  2. How would you simplify a fraction involving higher powers of 2?
  3. Can this problem be interpreted in terms of functions?
  4. What would happen if nn were a specific number, like n=2n = 2?
  5. Are there other methods to simplify expressions with exponential terms?

Tip: Always double-check the expression for any possible errors or misinterpretations when the provided answer choices don’t match.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Simplifying Expressions

Formulas

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Theorems

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Suitable Grade Level

Advanced High School