Question
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified form of the expression is \( \frac{5}{27} \).
Solution
To simplify the expression:
\[
\frac{5^{2n+1} \cdot 3^{2n-3}}{15^{2n}}
\]
**Step 1: Break Down the Denominator**
First, express \(15^{2n}\) in terms of its prime factors:
\[
15^{2n} = (3 \cdot 5)^{2n} = 3^{2n} \cdot 5^{2n}
\]
**Step 2: Rewrite the Original Expression**
Substitute the expression for \(15^{2n}\) back into the original fraction:
\[
\frac{5^{2n+1} \cdot 3^{2n-3}}{3^{2n} \cdot 5^{2n}}
\]
**Step 3: Simplify the Fraction**
Simplify by subtracting the exponents of like bases:
\[
= 5^{(2n+1) - 2n} \cdot 3^{(2n-3) - 2n}
\]
\[
= 5^{1} \cdot 3^{-3}
\]
**Step 4: Simplify Further**
Recall that \(3^{-3} = \frac{1}{3^3} = \frac{1}{27}\). Therefore:
\[
5^{1} \cdot 3^{-3} = 5 \cdot \frac{1}{27} = \frac{5}{27}
\]
**Final Answer:**
\[
\frac{5}{27}
\]
Reviewed and approved by the UpStudy tutoring team
Like
error msg
Explain 
Simplify this solution Beyond the Answer
Let's simplify the expression \( \frac{5^{2n+1} \cdot 3^{2n-3}}{15^{2n}} \). First, observe that \( 15 \) can be expressed as \( 3 \cdot 5 \). Thus, we have: \[ 15^{2n} = (3 \cdot 5)^{2n} = 3^{2n} \cdot 5^{2n} \] Now, we can rewrite the original expression by substituting \( 15^{2n} \): \[ \frac{5^{2n+1} \cdot 3^{2n-3}}{15^{2n}} = \frac{5^{2n+1} \cdot 3^{2n-3}}{3^{2n} \cdot 5^{2n}} \] Now, we can simplify this expression by separating the fractions: \[ = \frac{5^{2n+1}}{5^{2n}} \cdot \frac{3^{2n-3}}{3^{2n}} \] This results in: \[ = 5^{(2n+1) - 2n} \cdot 3^{(2n-3) - 2n} = 5^{1} \cdot 3^{-3} = \frac{5}{3^{3}} = \frac{5}{27} \] So, the final answer is: \[ \frac{5}{27} \]
