(3m^2n^7/m)^5

3 min read Jun 16, 2024
(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:

  1. Distribute the exponent: (3m^2n^7/m)^5 = 3^5 * (m^2)^5 * (n^7)^5 / (m)^5

  2. Simplify each term: 3^5 * (m^2)^5 * (n^7)^5 / (m)^5 = 243 * m^(25) * n^(75) / m^5

  3. Simplify further: 243 * m^(25) * n^(75) / m^5 = 243 * m^10 * n^35 / m^5

  4. Apply the quotient rule: 243 * m^10 * n^35 / m^5 = 243 * m^(10-5) * n^35

  5. 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.

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