(6x^-3y^5/2xy^2z^6)^5

Jasonbradley

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Jun 16, 2024

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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

1. 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

2. 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

3. Simplify the exponents:

   • Multiply the exponents: = 243x^-20y^15z^-30

4. 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.