To simplify the given expression [tex]\(\frac{x^{-6}}{x^{-4}}\)[/tex] and express the answer using positive exponents only, follow these steps:
1. Understand the properties of exponents:
- When dividing like bases, you subtract the exponent of the denominator from the exponent of the numerator: [tex]\[ \frac{a^m}{a^n} = a^{m-n} \][/tex]
2. Apply this property to the given expression:
[tex]\[ \frac{x^{-6}}{x^{-4}} = x^{-6 - (-4)} \][/tex]
3. Simplify the exponent:
- Subtracting a negative is equivalent to adding the positive value:
[tex]\[ x^{-6 - (-4)} = x^{-6 + 4} \][/tex]
[tex]\[ x^{-6 + 4} = x^{-2} \][/tex]
4. Express the answer using positive exponents:
- Recall that [tex]\(x^{-n} = \frac{1}{x^n}\)[/tex]:
[tex]\[ x^{-2} = \frac{1}{x^2} \][/tex]
Therefore, the simplified expression using positive exponents is:
[tex]\[ \frac{x^{-6}}{x^{-4}} = \frac{1}{x^2} \][/tex]
Alternative way to express the answer using exponential notation:
- The expression [tex]\(x^{-2}\)[/tex] can also be written as an exponent in terms of the natural logarithm's properties:
[tex]\[ x^{-2} = \exp(-2 \log(x)) \][/tex]
Thus, the complete transformation of the initial expression is:
[tex]\[ \frac{x^{-6}}{x^{-4}} = x^{-2} = \frac{1}{x^2} = \exp(-2 \log(x)) \][/tex]
