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Simplifying Expressions with Exponents: (-6x^3y^5)^4
This article will explore the simplification of the expression (-6x^3y^5)^4. Understanding how to work with exponents and apply the rules of exponents is crucial in algebra and various scientific disciplines.
Key Concepts:
- Exponents: An exponent indicates the number of times a base is multiplied by itself. In the expression (-6x^3y^5)^4, 4 is the exponent and (-6x^3y^5) is the base.
- Power of a Product: When raising a product to a power, we raise each factor within the product to that power. This is represented by the rule (ab)^n = a^n * b^n.
- Power of a Power: When raising a power to another power, we multiply the exponents. This rule is represented by (a^m)^n = a^(m*n).
Simplifying the Expression:
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Apply the Power of a Product rule: (-6x^3y^5)^4 = (-6)^4 * (x^3)^4 * (y^5)^4
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Apply the Power of a Power rule: (-6)^4 * (x^3)^4 * (y^5)^4 = (-6)^4 * x^(34) * y^(54)
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Calculate the exponents: (-6)^4 * x^(34) * y^(54) = 1296 * x^12 * y^20
Therefore, the simplified form of (-6x^3y^5)^4 is 1296x^12y^20.
Summary:
By utilizing the rules of exponents, we successfully simplified the complex expression (-6x^3y^5)^4. Understanding and applying these rules is vital in simplifying more intricate expressions and performing various mathematical operations.