https://r2.erweima.ai/mathbot/image/17259211451283u344va5.jpg

Let's solve each part of the problem by applying exponentiation properties to express the terms in the form \( k \cdot a^n \).

### Part (a):
We have the expression:

\[
\frac{2 \cdot 5^n}{3^{n+1}}
\]

We can rewrite the denominator as follows:

\[
3^{n+1} = 3^n \cdot 3
\]

Thus, the expression becomes:

\[
\frac{2 \cdot 5^n}{3^n \cdot 3} = 2 \cdot \frac{5^n}{3^n} \cdot \frac{1}{3}
\]

Now, using the property \( \frac{a^n}{b^n} = \left( \frac{a}{b} \right)^n \), we get:

\[
2 \cdot \left( \frac{5}{3} \right)^n \cdot \frac{1}{3}
\]

Finally, simplifying:

\[
\frac{2}{3} \cdot \left( \frac{5}{3} \right)^n
\]

Thus, part (a) can be expressed as:

\[
\boxed{\frac{2}{3} \cdot \left( \frac{5}{3} \right)^n}
\]

### Part (b):
We have the expression:

\[
\frac{4^{n+1}}{2^{n-2}}
\]

We can rewrite the numerator as follows:

\[
4^{n+1} = (2^2)^{n+1} = 2^{2(n+1)} = 2^{2n+2}
\]

Thus, the expression becomes:

\[
\frac{2^{2n+2}}{2^{n-2}} = 2^{(2n+2) - (n-2)} = 2^{2n+2 - n + 2} = 2^{n+4}
\]

Thus, part (b) simplifies to:

\[
\boxed{2^{n+4}}
\]

### Final Answers:
- (a) \( \frac{2}{3} \cdot \left( \frac{5}{3} \right)^n \)
- (b) \( 2^{n+4} \)

Would you like further details on any of the steps? Here are five related questions to explore this topic further:

1. How can you apply the property \( \frac{a^n}{b^n} = \left( \frac{a}{b} \right)^n \) in different scenarios?
2. How do you simplify expressions involving sums and differences of exponents?
3. Can you express \( k \cdot a^n \) in different equivalent forms by varying the base \( a \)?
4. How do you handle negative exponents in similar problems?
5. What are other common properties of exponentiation that are useful in solving equations?

**Tip:** When simplifying exponential expressions, always try to factor common terms in the numerator and denominator for easier manipulation.

Applying properties of exponentiation, express in the form k.a^n:

Explore the detailed step-by-step method to express exponential expressions in the form k.a^n using properties of exponentiation. Suitable for high school students learning advanced algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation