Answer :
To determine which expressions are equivalent to \( 5^4 \cdot 5^x \), let's analyze each option step-by-step:
### Step-by-Step Analysis
1. Option A: \( 5^{4+x} \)
- Property of Exponents: When multiplying like bases, you add the exponents: \( a^m \cdot a^n = a^{m+n} \).
- Applying this property:
[tex]\[ 5^4 \cdot 5^x = 5^{4+x} \][/tex]
- This expression is correct.
2. Option B: \( 625 \cdot 5^x \)
- Simplifying Base: Recognize that \( 625 \) can be written as \( 5^4 \) (since \( 5^4 = 625 \)).
- Rewriting the expression:
[tex]\[ 625 \cdot 5^x = 5^4 \cdot 5^x \][/tex]
- Based on the property of exponents, it simplifies to \( 5^{4+x} \), which is the original expression. Thus, this expression is correct.
3. Option C: \( 25^{4x} \)
- Rewriting Base: Notice that \( 25 \) can be written as \( 5^2 \) (since \( 25 = 5^2 \)).
- Rewriting the expression:
[tex]\[ 25^{4x} = (5^2)^{4x} = 5^{2 \cdot 4x} = 5^{8x} \][/tex]
- This is different from \( 5^{4+x} \). Therefore, this expression is not correct.
4. Option D: \( (5 \cdot x)^4 \)
- Expanding inside the parenthesis:
[tex]\[ (5 \cdot x)^4 = 5^4 \cdot x^4 \][/tex]
- This expression includes \( x^4 \), which is not present in the original \( 5^4 \cdot 5^x \). Therefore, this expression is not correct.
5. Option E: \( 5^{4-x} \)
- This expression involves subtraction in the exponent:
[tex]\[ 5^{4-x} \][/tex]
- This is not the same as \( 5^{4+x} \). Therefore, this expression is not correct.
6. Option F: \( 5^{4x} \)
- This expression uses multiplication within the exponent:
[tex]\[ 5^{4x} \][/tex]
- This is different from \( 5^{4+x} \). Therefore, this expression is not correct.
### Conclusion
Based on our analysis, the expressions equivalent to \( 5^4 \cdot 5^x \) are:
- \( 5^{4+x} \) (Option A)
- \( 625 \cdot 5^x \) (Option B)
Therefore, the correct options are A and B (or options 1 and 2).
### Step-by-Step Analysis
1. Option A: \( 5^{4+x} \)
- Property of Exponents: When multiplying like bases, you add the exponents: \( a^m \cdot a^n = a^{m+n} \).
- Applying this property:
[tex]\[ 5^4 \cdot 5^x = 5^{4+x} \][/tex]
- This expression is correct.
2. Option B: \( 625 \cdot 5^x \)
- Simplifying Base: Recognize that \( 625 \) can be written as \( 5^4 \) (since \( 5^4 = 625 \)).
- Rewriting the expression:
[tex]\[ 625 \cdot 5^x = 5^4 \cdot 5^x \][/tex]
- Based on the property of exponents, it simplifies to \( 5^{4+x} \), which is the original expression. Thus, this expression is correct.
3. Option C: \( 25^{4x} \)
- Rewriting Base: Notice that \( 25 \) can be written as \( 5^2 \) (since \( 25 = 5^2 \)).
- Rewriting the expression:
[tex]\[ 25^{4x} = (5^2)^{4x} = 5^{2 \cdot 4x} = 5^{8x} \][/tex]
- This is different from \( 5^{4+x} \). Therefore, this expression is not correct.
4. Option D: \( (5 \cdot x)^4 \)
- Expanding inside the parenthesis:
[tex]\[ (5 \cdot x)^4 = 5^4 \cdot x^4 \][/tex]
- This expression includes \( x^4 \), which is not present in the original \( 5^4 \cdot 5^x \). Therefore, this expression is not correct.
5. Option E: \( 5^{4-x} \)
- This expression involves subtraction in the exponent:
[tex]\[ 5^{4-x} \][/tex]
- This is not the same as \( 5^{4+x} \). Therefore, this expression is not correct.
6. Option F: \( 5^{4x} \)
- This expression uses multiplication within the exponent:
[tex]\[ 5^{4x} \][/tex]
- This is different from \( 5^{4+x} \). Therefore, this expression is not correct.
### Conclusion
Based on our analysis, the expressions equivalent to \( 5^4 \cdot 5^x \) are:
- \( 5^{4+x} \) (Option A)
- \( 625 \cdot 5^x \) (Option B)
Therefore, the correct options are A and B (or options 1 and 2).
