Answer :
Certainly! To determine the correct way to rewrite \( p^m p^n \), let's go through the options step-by-step and analyze each one based on the properties of exponents.
1. Option \( p^{(m+n)} \)
- When you multiply two expressions with the same base, you add their exponents. This is one of the fundamental properties of exponents.
- Thus, \( p^m \cdot p^n = p^{m+n} \).
2. Option \( p^{m n} \)
- This suggests that the exponents are being multiplied rather than added. However, this is not consistent with the rules of exponents regarding multiplication.
- The rule states that the exponents should be added, not multiplied. Therefore, \( p^m \cdot p^n \neq p^{m n} \).
3. Option \( p^2 m n \)
- This option changes the base \( p \) and introduces multiplicative terms \( m \) and \( n \) in a manner that does not follow the rules of exponents.
- Such a representation does not logically follow from \( p^m \cdot p^n \). Therefore, \( p^m \cdot p^n \neq p^2 m n \).
4. Option \( 2 p^{(m+n)} \)
- In this option, there is an extraneous factor of \( 2 \) in front of \( p^{(m+n)} \).
- The correct application of exponent rules does not introduce any additional multiplicative constants; therefore, \( p^m \cdot p^n \neq 2 p^{(m+n)} \).
Based on the above analysis, the correct way to rewrite \( p^m p^n \) is
[tex]\[ p^{(m+n)} \][/tex]
Thus, the correct option is the first one:
[tex]\[ \boxed{1} \][/tex]
1. Option \( p^{(m+n)} \)
- When you multiply two expressions with the same base, you add their exponents. This is one of the fundamental properties of exponents.
- Thus, \( p^m \cdot p^n = p^{m+n} \).
2. Option \( p^{m n} \)
- This suggests that the exponents are being multiplied rather than added. However, this is not consistent with the rules of exponents regarding multiplication.
- The rule states that the exponents should be added, not multiplied. Therefore, \( p^m \cdot p^n \neq p^{m n} \).
3. Option \( p^2 m n \)
- This option changes the base \( p \) and introduces multiplicative terms \( m \) and \( n \) in a manner that does not follow the rules of exponents.
- Such a representation does not logically follow from \( p^m \cdot p^n \). Therefore, \( p^m \cdot p^n \neq p^2 m n \).
4. Option \( 2 p^{(m+n)} \)
- In this option, there is an extraneous factor of \( 2 \) in front of \( p^{(m+n)} \).
- The correct application of exponent rules does not introduce any additional multiplicative constants; therefore, \( p^m \cdot p^n \neq 2 p^{(m+n)} \).
Based on the above analysis, the correct way to rewrite \( p^m p^n \) is
[tex]\[ p^{(m+n)} \][/tex]
Thus, the correct option is the first one:
[tex]\[ \boxed{1} \][/tex]
