%5E-x%5E2-81%2F256.jpeg "(3/4)^-x^2 81/256")
Simplifying Exponential Expressions: (3/4)^-x^2 * 81/256
This article will walk you through the process of simplifying the expression (3/4)^-x^2 * 81/256. We'll utilize the properties of exponents to achieve a concise and understandable solution.
Understanding the Properties
Before diving into the simplification, let's recall some key properties of exponents:
- Negative exponents: a^-n = 1/a^n
- Fractional exponents: a^(m/n) = (a^m)^(1/n) = (a^(1/n))^m
- Product of powers: a^m * a^n = a^(m+n)
Simplifying the Expression
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Address the negative exponent: (3/4)^-x^2 = 1 / (3/4)^x^2
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Rewrite the fraction 81/256 in terms of 3/4: 81/256 = (3^4) / (4^4) = (3/4)^4
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Combine the terms: 1 / (3/4)^x^2 * (3/4)^4 = (3/4)^4 / (3/4)^x^2
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Apply the product of powers rule: (3/4)^4 / (3/4)^x^2 = (3/4)^(4 - x^2)
Final Result
The simplified expression is (3/4)^(4 - x^2).
Conclusion
By applying the fundamental properties of exponents, we were able to simplify the expression into a more manageable form. This approach can be applied to various similar expressions involving negative exponents and fractions.