Math Problem Statement

Simplify. Express your answer using exponents. (3m^9n^3)^2

Solution

The expression you provided is:

(3m9n3)2(3m^9n^3)^2

To simplify this, apply the power of a product rule: (ab)n=anbn(ab)^n = a^n \cdot b^n.

Step-by-step:

  1. First, apply the exponent to each part inside the parentheses:

(3m9n3)2=32(m9)2(n3)2(3m^9n^3)^2 = 3^2 \cdot (m^9)^2 \cdot (n^3)^2

  1. Simplify each part:
    • 32=93^2 = 9
    • (m9)2=m9×2=m18(m^9)^2 = m^{9 \times 2} = m^{18}
    • (n3)2=n3×2=n6(n^3)^2 = n^{3 \times 2} = n^6

So, the simplified expression is:

9m18n69m^{18}n^6

Would you like a more detailed explanation of any part of the solution?

Here are some related questions to explore further:

  1. What is the difference between the product and power rules in exponents?
  2. How do you simplify expressions with negative exponents?
  3. What happens if you multiply two expressions with the same base but different exponents?
  4. How can you handle expressions with fractional exponents?
  5. Can exponents be applied to terms with variables raised to different powers?

Tip: When simplifying expressions with exponents, always be mindful of the rules of exponents, such as the power of a product, power of a power, and the product of powers.

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Math Problem Analysis

Mathematical Concepts

Exponents
Algebra
Exponentiation Rules

Formulas

(ab)^n = a^n * b^n
(x^m)^n = x^(m*n)

Theorems

Exponentiation Rules

Suitable Grade Level

Grades 7-9