%5E4.jpeg "(3x^3y^5)^4")
Simplifying Exponential Expressions: (3x^3y^5)^4
In mathematics, simplifying expressions is a crucial step in solving equations and understanding their meaning. One common type of expression involves raising a term with multiple variables and coefficients to a power. This article will guide you through the process of simplifying the expression (3x^3y^5)^4.
Understanding the Rules
Before diving into the simplification, let's recall some fundamental rules of exponents:
- Power of a product: (ab)^n = a^n * b^n
- Power of a power: (a^m)^n = a^(m*n)
Applying the Rules
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Distribute the exponent: Using the power of a product rule, we distribute the exponent '4' to each factor inside the parentheses: (3x^3y^5)^4 = 3^4 * (x^3)^4 * (y^5)^4
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Simplify the exponents: Applying the power of a power rule, we multiply the exponents: 3^4 * (x^3)^4 * (y^5)^4 = 3^4 * x^(34) * y^(54)
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Calculate the results: 3^4 * x^(34) * y^(54) = 81 * x^12 * y^20
Final Result
Therefore, the simplified form of (3x^3y^5)^4 is 81x^12y^20.
Key Takeaways
- Remember the rules of exponents for simplifying expressions.
- Distribute the exponent to all factors inside the parentheses.
- Multiply the exponents when raising a power to another power.
- Simplify the expression by calculating the numerical values and combining terms.