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Simplifying the Expression: (25x^3y^4)^1/2
In mathematics, simplifying expressions is a fundamental skill. Let's explore how to simplify the expression (25x^3y^4)^1/2.
Understanding the Properties of Exponents
The expression involves exponents and fractional powers. Here are some key properties we'll use:
- Product of Powers: (a^m)^n = a^(m*n)
- Fractional Exponent: a^(1/n) = √n(a)
Applying the Properties
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Distribute the exponent: We apply the product of powers rule to distribute the 1/2 exponent to each factor inside the parentheses: (25x^3y^4)^1/2 = 25^(1/2) * (x^3)^(1/2) * (y^4)^(1/2)
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Simplify each factor:
- 25^(1/2) = √25 = 5
- (x^3)^(1/2) = x^(3/2)
- (y^4)^(1/2) = y^(4/2) = y^2
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Combine the terms: 5 * x^(3/2) * y^2
Therefore, the simplified form of (25x^3y^4)^1/2 is 5x^(3/2)y^2.
Additional Notes
- The expression can be further simplified if we express x^(3/2) as √x^3.
- The simplified expression represents the square root of the original expression.
- Understanding these properties of exponents allows us to manipulate and simplify expressions efficiently.