Question
Fully simplify using only positive exponents. \[ \frac{25 x^{2} y^{8}}{25 x^{7} y^{5}} \]
Ask by Boone Pope. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\frac{y^{3}}{x^{5}}
\]
Solution
To simplify the expression \(\frac{25 x^{2} y^{8}}{25 x^{7} y^{5}}\) using only positive exponents, follow these steps:
1. **Simplify the coefficients:**
\[
\frac{25}{25} = 1
\]
2. **Simplify the \(x\) terms using the laws of exponents (\( \frac{x^a}{x^b} = x^{a-b} \)):**
\[
\frac{x^2}{x^7} = x^{2-7} = x^{-5}
\]
To express this with a positive exponent:
\[
x^{-5} = \frac{1}{x^5}
\]
3. **Simplify the \(y\) terms using the laws of exponents:**
\[
\frac{y^8}{y^5} = y^{8-5} = y^3
\]
4. **Combine the simplified parts:**
\[
1 \times \frac{y^3}{x^5} = \frac{y^3}{x^5}
\]
**Final Simplified Expression:**
\[
\frac{y^{3}}{x^{5}}
\]
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Simplify this solution The Deep Dive
To simplify the expression \(\frac{25 x^{2} y^{8}}{25 x^{7} y^{5}}\), we start by canceling the common factor of 25 in the numerator and denominator, leaving us with: \[ \frac{x^{2} y^{8}}{x^{7} y^{5}}. \] Now, we can simplify the \(x\) terms and the \(y\) terms separately using the exponent rules. For the \(x\) terms: \[ \frac{x^{2}}{x^{7}} = x^{2-7} = x^{-5}. \] For the \(y\) terms: \[ \frac{y^{8}}{y^{5}} = y^{8-5} = y^{3}. \] Combining these results, we get: \[ x^{-5} y^{3}. \] To express this with only positive exponents, we can rewrite \(x^{-5}\) as \(\frac{y^{3}}{x^{5}}\). Thus, the fully simplified expression is: \[ \frac{y^{3}}{x^{5}}. \]
