Answer :
Certainly! Let's break down the expression \( x^{-4} \) and analyze the options given to identify the expression that is equivalent to it.
### Step-by-Step Solution
1. Understanding \( x^{-4} \):
- The expression \( x^{-4} \) involves a negative exponent. A negative exponent indicates that the base is on the wrong side of a fraction line. So, \( x^{-4} \) can be rewritten in positive exponential form as:
[tex]\[ x^{-4} = \frac{1}{x^4} \][/tex]
2. Analyzing the Options:
- Option 1: \( x^{-1} + x^{-3} \)
[tex]\[ x^{-1} + x^{-3} = \frac{1}{x} + \frac{1}{x^3} \][/tex]
This expression involves two separate terms with different exponents and does not combine to \( x^{-4} \), so it is not equivalent.
- Option 2: \( x^{-2} \cdot x^{-2} \)
[tex]\[ x^{-2} \cdot x^{-2} = x^{(-2) + (-2)} = x^{-4} \][/tex]
This expression combines using properties of exponents (where exponents add when the same bases are multiplied) and is directly equivalent to \( x^{-4} \).
- Option 3: \( x^2 - x^6 \)
[tex]\[ x^2 - x^6 \][/tex]
This expression is a subtraction of two terms with positive exponents and does not represent \( x^{-4} \).
- Option 4: \( x^{-1} \cdot x^4 \)
[tex]\[ x^{-1} \cdot x^4 = x^{(-1) + 4} = x^3 \][/tex]
This expression simplifies to \( x^3 \), which is not equivalent to \( x^{-4} \).
### Conclusion
Based on the analysis of each option and properties of exponents, the expression that is equivalent to \( x^{-4} \) is:
[tex]\[ \boxed{x^{-2} \cdot x^{-2}} \][/tex]
### Step-by-Step Solution
1. Understanding \( x^{-4} \):
- The expression \( x^{-4} \) involves a negative exponent. A negative exponent indicates that the base is on the wrong side of a fraction line. So, \( x^{-4} \) can be rewritten in positive exponential form as:
[tex]\[ x^{-4} = \frac{1}{x^4} \][/tex]
2. Analyzing the Options:
- Option 1: \( x^{-1} + x^{-3} \)
[tex]\[ x^{-1} + x^{-3} = \frac{1}{x} + \frac{1}{x^3} \][/tex]
This expression involves two separate terms with different exponents and does not combine to \( x^{-4} \), so it is not equivalent.
- Option 2: \( x^{-2} \cdot x^{-2} \)
[tex]\[ x^{-2} \cdot x^{-2} = x^{(-2) + (-2)} = x^{-4} \][/tex]
This expression combines using properties of exponents (where exponents add when the same bases are multiplied) and is directly equivalent to \( x^{-4} \).
- Option 3: \( x^2 - x^6 \)
[tex]\[ x^2 - x^6 \][/tex]
This expression is a subtraction of two terms with positive exponents and does not represent \( x^{-4} \).
- Option 4: \( x^{-1} \cdot x^4 \)
[tex]\[ x^{-1} \cdot x^4 = x^{(-1) + 4} = x^3 \][/tex]
This expression simplifies to \( x^3 \), which is not equivalent to \( x^{-4} \).
### Conclusion
Based on the analysis of each option and properties of exponents, the expression that is equivalent to \( x^{-4} \) is:
[tex]\[ \boxed{x^{-2} \cdot x^{-2}} \][/tex]
