%5E3(2mn)%5E-1(8n%5E3)%5E2%2F3.jpeg "(3m^2)^3(2mn)^-1(8n^3)^2/3")
Simplifying Algebraic Expressions: A Step-by-Step Guide
This article will guide you through the simplification of the algebraic expression:
(3m^2)^3(2mn)^-1(8n^3)^2/3
Let's break down the process step-by-step:
1. Understanding the Properties of Exponents
Before we begin simplification, let's recall some key exponent properties:
- Product of Powers: x^m * x^n = x^(m+n)
- Power of a Power: (x^m)^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
- Negative Exponent: x^-n = 1/x^n
2. Simplifying Each Term
a. (3m^2)^3
Using the power of a power property: (3m^2)^3 = 3^3 * (m^2)^3 = 27m^6
b. (2mn)^-1
Using the negative exponent property: (2mn)^-1 = 1/(2mn)^1 = 1/(2mn)
c. (8n^3)^2/3
Using the power of a power property: (8n^3)^2/3 = 8^(2/3) * (n^3)^(2/3) = 4n^2
3. Combining the Simplified Terms
Now we have: 27m^6 * (1/(2mn)) * 4n^2
4. Simplifying Further
- Multiplying the coefficients: 27 * (1/2) * 4 = 54
- Combining the variables using the product of powers property: m^6 * n^2 / mn = m^5 * n
5. The Final Simplified Expression
The fully simplified expression is 54m^5n.
Conclusion
By understanding and applying the basic properties of exponents, we have successfully simplified the given algebraic expression. Remember to take each step carefully and be mindful of the order of operations.