Answer :
To simplify the expression \(\frac{16 r^5 r^3}{8 r^2 s^6}\), we will simplify it step by step.
1. Combine the powers of \( r \) in the numerator:
- The expression \( r^5 r^3 \) means \( r \) raised to the power of \( 5 \) multiplied by \( r \) raised to the power of \( 3 \).
- According to the rule of exponents, \( a^m a^n = a^{m + n} \). Therefore, \( r^5 r^3 = r^{5 + 3} = r^8 \).
The expression now becomes:
[tex]\[ \frac{16 r^8}{8 r^2 s^6} \][/tex]
2. Simplify the constant term:
- Divide the constants in the numerator and denominator: \(\frac{16}{8} = 2\).
The expression now simplifies to:
[tex]\[ \frac{2 r^8}{r^2 s^6} \][/tex]
3. Simplify the \( r \) term:
- Use the rule of exponents for division: \(\frac{a^m}{a^n} = a^{m-n}\).
- Applying this to \( r \):
[tex]\[ \frac{r^8}{r^2} = r^{8-2} = r^6 \][/tex]
So the expression now simplifies further to:
[tex]\[ \frac{2 r^6}{s^6} \][/tex]
The fully simplified expression is:
[tex]\[ \frac{2 r^6}{s^6} \][/tex]
Given the options, the correct match for our simplified expression is:
- None of the provided options exactly match \(\frac{2 r^6}{s^6}\).
However, the provided solution gives me the closest match:
B. \(\frac{2 r^4}{s^4}\)
Based on derivation, there was possibly a typographical or interpretative error in the Python code reasoning since:
1. We shouldn't decrement s terms.
2. \(2r^6\) appeared only via correct solving.
Answer: the given correct answer is; none, otherwise recheck the values presented.
1. Combine the powers of \( r \) in the numerator:
- The expression \( r^5 r^3 \) means \( r \) raised to the power of \( 5 \) multiplied by \( r \) raised to the power of \( 3 \).
- According to the rule of exponents, \( a^m a^n = a^{m + n} \). Therefore, \( r^5 r^3 = r^{5 + 3} = r^8 \).
The expression now becomes:
[tex]\[ \frac{16 r^8}{8 r^2 s^6} \][/tex]
2. Simplify the constant term:
- Divide the constants in the numerator and denominator: \(\frac{16}{8} = 2\).
The expression now simplifies to:
[tex]\[ \frac{2 r^8}{r^2 s^6} \][/tex]
3. Simplify the \( r \) term:
- Use the rule of exponents for division: \(\frac{a^m}{a^n} = a^{m-n}\).
- Applying this to \( r \):
[tex]\[ \frac{r^8}{r^2} = r^{8-2} = r^6 \][/tex]
So the expression now simplifies further to:
[tex]\[ \frac{2 r^6}{s^6} \][/tex]
The fully simplified expression is:
[tex]\[ \frac{2 r^6}{s^6} \][/tex]
Given the options, the correct match for our simplified expression is:
- None of the provided options exactly match \(\frac{2 r^6}{s^6}\).
However, the provided solution gives me the closest match:
B. \(\frac{2 r^4}{s^4}\)
Based on derivation, there was possibly a typographical or interpretative error in the Python code reasoning since:
1. We shouldn't decrement s terms.
2. \(2r^6\) appeared only via correct solving.
Answer: the given correct answer is; none, otherwise recheck the values presented.
