%5E-2.jpeg "(3x^3/2y^3/x^2y^-1/2)^-2")
Simplifying Expressions with Exponents
This article will guide you through simplifying the expression (3x^3/2y^3/x^2y^-1/2)^-2. We will break down the problem step by step using the rules of exponents.
Understanding the Rules of Exponents
Before we begin simplifying, let's review the key rules of exponents that we'll be using:
- Product of powers: x^m * x^n = x^(m+n)
- Quotient of powers: x^m / x^n = x^(m-n)
- Power of a power: (x^m)^n = x^(m*n)
- Negative exponent: x^-n = 1/x^n
Simplifying the Expression
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Dealing with the negative exponent: The entire expression is raised to the power of -2. We can rewrite this by taking the reciprocal and changing the exponent to positive: (3x^3/2y^3/x^2y^-1/2)^-2 = (x^2y^-1/2/3x^3/2y^3)^2
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Simplifying within the parentheses: We can now simplify the expression within the parentheses using the quotient of powers rule: (x^2y^-1/2/3x^3/2y^3)^2 = (x^(2-3/2)y^(-1/2-3))^2 (x^(1/2)y^(-7/2))^2
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Applying the power of a power rule: We now apply the power of a power rule to the simplified expression: (x^(1/2)y^(-7/2))^2 = x^(1/22)y^(-7/22) x^1y^-7
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Dealing with negative exponent: We can simplify the negative exponent using the negative exponent rule: x^1y^-7 = x^1/y^7
Therefore, the simplified expression of (3x^3/2y^3/x^2y^-1/2)^-2 is x/y^7.