%5E5.jpeg "(3m^2n^7/m)^5")
Simplifying Exponential Expressions: (3m^2n^7/m)^5
This article will guide you through simplifying the expression (3m^2n^7/m)^5. We'll use the rules of exponents to break down the expression and arrive at a simplified form.
Understanding the Rules
Before we begin, let's review some key exponent rules:
- Product of Powers: x^m * x^n = x^(m+n)
- Quotient of Powers: x^m / x^n = x^(m-n)
- Power of a Power: (x^m)^n = x^(m*n)
- Power of a Product: (x*y)^n = x^n * y^n
- Power of a Quotient: (x/y)^n = x^n / y^n
Applying the Rules
Let's apply these rules to simplify our expression:
-
Distribute the exponent: (3m^2n^7/m)^5 = 3^5 * (m^2)^5 * (n^7)^5 / (m)^5
-
Simplify each term: 3^5 * (m^2)^5 * (n^7)^5 / (m)^5 = 243 * m^(25) * n^(75) / m^5
-
Simplify further: 243 * m^(25) * n^(75) / m^5 = 243 * m^10 * n^35 / m^5
-
Apply the quotient rule: 243 * m^10 * n^35 / m^5 = 243 * m^(10-5) * n^35
-
Final simplification: 243 * m^(10-5) * n^35 = 243m^5n^35
Conclusion
By applying the rules of exponents, we have simplified the expression (3m^2n^7/m)^5 to 243m^5n^35. Remember, understanding these rules is crucial for working with exponential expressions, which appear frequently in various fields of mathematics and science.