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 stepbystep 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^(mn)
 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

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

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

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

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.