Math Problem Statement

Simplify \(\frac{3^{n+3} - 3 \cdot 3^n}{4 \cdot 3^{n+2}}\)

Solution

The expression to simplify is:

3n+333n43n+2\frac{3^{n+3} - 3 \cdot 3^n}{4 \cdot 3^{n+2}}

Let's simplify this step-by-step.

Step 1: Simplify the numerator

Rewrite the numerator by factoring out 3n3^n: 3n+333n=3n3333n=3n(273)=3n243^{n+3} - 3 \cdot 3^n = 3^n \cdot 3^3 - 3 \cdot 3^n = 3^n (27 - 3) = 3^n \cdot 24

So, the numerator becomes 243n24 \cdot 3^n.

Step 2: Rewrite the expression

Now, substitute the simplified numerator back: 243n43n+2\frac{24 \cdot 3^n}{4 \cdot 3^{n+2}}

Step 3: Simplify the denominator

Rewrite 3n+23^{n+2} as 3n323^n \cdot 3^2: 243n43n9\frac{24 \cdot 3^n}{4 \cdot 3^n \cdot 9}

Step 4: Cancel 3n3^n from the numerator and the denominator

This gives: 2449\frac{24}{4 \cdot 9}

Step 5: Simplify the constants

Calculate 49=364 \cdot 9 = 36, so the expression becomes: 2436=23\frac{24}{36} = \frac{2}{3}

Final Answer:

23\frac{2}{3}

Would you like further clarification on any steps?

Here are some related questions for practice:

  1. How would you simplify 5n+455n105n+2\frac{5^{n+4} - 5 \cdot 5^n}{10 \cdot 5^{n+2}}?
  2. What are the steps to simplify 2n+522n82n+3\frac{2^{n+5} - 2 \cdot 2^n}{8 \cdot 2^{n+3}}?
  3. How do you simplify expressions with exponents in both the numerator and the denominator?
  4. How would you factor out terms in expressions like xm+2xxmx^{m+2} - x \cdot x^m?
  5. How would the simplification differ if the base in the exponents was negative?

Tip: When simplifying expressions with exponents, factor out common terms to reduce the expression more efficiently.

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Math Problem Analysis

Mathematical Concepts

Exponents
Algebra
Simplification

Formulas

Power of a Product: \(a^{m+n} = a^m \cdot a^n\)
Factoring out common terms

Theorems

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Suitable Grade Level

Grades 9-10