%5E5.jpeg "(6x^-3y^5/2xy^2z^6)^5")
Simplifying Expressions with Exponents
This article will guide you through simplifying the expression (6x^-3y^5/2xy^2z^6)^5. We'll break down the process step-by-step using the rules of exponents.
Understanding the Properties of Exponents
Before we begin simplifying, let's recall the key properties of exponents:
- Product of Powers: x^m * x^n = x^(m+n)
- Quotient of Powers: x^m / x^n = x^(m-n)
- Power of a Product: (xy)^n = x^n * y^n
- Power of a Quotient: (x/y)^n = x^n / y^n
- Power of a Power: (x^m)^n = x^(m*n)
Simplifying the Expression
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Simplify inside the parentheses:
- Apply the quotient of powers rule to the variables: (6x^-3y^5/2xy^2z^6) = (6/2) * (x^-3/x) * (y^5/y^2) * (1/z^6)
- Simplify the coefficients and exponents: = 3x^-4y^3z^-6
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Apply the power of a power rule:
- Raise each term inside the parentheses to the power of 5: (3x^-4y^3z^-6)^5 = 3^5 * (x^-4)^5 * (y^3)^5 * (z^-6)^5
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Simplify the exponents:
- Multiply the exponents: = 243x^-20y^15z^-30
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Express with positive exponents:
- Move the terms with negative exponents to the denominator: = 243y^15 / x^20z^30
Final Answer
The simplified expression for (6x^-3y^5/2xy^2z^6)^5 is 243y^15 / x^20z^30.