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Simplifying Expressions with Fractional Exponents
This article will guide you through simplifying the expression (9x⁴y⁶)³/². We'll break down the process step by step, using the properties of exponents to achieve the simplest form.
Understanding the Properties of Exponents
Before we dive into the simplification, let's recall the key properties of exponents that we'll use:
- Product of powers: (xᵃ)ᵇ = xᵃᵇ
- Power of a product: (xy)ᵃ = xᵃyᵃ
- Fractional exponent: x¹/ⁿ = ⁿ√x
Simplifying the Expression
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Applying the power of a product property:
(9x⁴y⁶)³/² = 9³/² * (x⁴)³/² * (y⁶)³/²
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Applying the product of powers property:
9³/² * (x⁴)³/² * (y⁶)³/² = 9³⁺¹/² * x⁴⁺³/² * y⁶⁺³/²
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Simplifying exponents:
9³⁺¹/² * x⁴⁺³/² * y⁶⁺³/² = 9⁷/² * x¹¹/² * y⁹
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Expressing the fractional exponent as a radical:
9⁷/² * x¹¹/² * y⁹ = √(9⁷) * √(x¹¹) * y⁹
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Simplifying the radical:
√(9⁷) * √(x¹¹) * y⁹ = 9³√9 * x⁵√x * y⁹
Therefore, the simplified form of (9x⁴y⁶)³/² is 9³√9 * x⁵√x * y⁹.
Conclusion
By applying the properties of exponents and simplifying radicals, we were able to express the original expression in its simplest form. Understanding these properties allows you to manipulate and simplify various expressions involving exponents with ease.