Math Problem Statement

Simplify the expression (-3x^4y^3)^4.

Solution

The expression shown in the image is:

(3x4y3)4(-3x^4y^3)^4

Let's simplify this step by step:

Step 1: Apply the power rule

Using the rule (ab)n=anbn(a \cdot b)^n = a^n \cdot b^n, we distribute the exponent 44 to each term inside the parentheses:

(3)4(x4)4(y3)4(-3)^4 \cdot (x^4)^4 \cdot (y^3)^4

Step 2: Simplify each term

  1. (3)4=81(-3)^4 = 81 (since (3)(-3) raised to an even power becomes positive).
  2. (x4)4=x44=x16(x^4)^4 = x^{4 \cdot 4} = x^{16} (using the power of a power rule: (am)n=amn(a^m)^n = a^{m \cdot n}).
  3. (y3)4=y34=y12(y^3)^4 = y^{3 \cdot 4} = y^{12}.

Step 3: Combine the results

(3x4y3)4=81x16y12(-3x^4y^3)^4 = 81x^{16}y^{12}

Final Answer:

81x16y12\boxed{81x^{16}y^{12}}

Would you like further explanation or related examples? Let me know!


Here are 5 related questions to expand your understanding:

  1. How does the power rule work for negative bases raised to even and odd powers?
  2. What happens if there’s an additional variable, such as (3x4y3z)4(-3x^4y^3z)^4?
  3. How do fractional exponents affect similar expressions?
  4. Can you explain how to simplify expressions like (2x5y2)3(2x^5y^{-2})^3?
  5. How do negative exponents influence the simplification process?

Tip: Always distribute exponents carefully to each part of the expression, especially when there’s a mix of coefficients and variables!

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Math Problem Analysis

Mathematical Concepts

Exponent Rules
Algebraic Simplification
Power of a Product Rule
Power of a Power Rule

Formulas

(a * b)^n = a^n * b^n
(a^m)^n = a^{m*n}

Theorems

Power Rule of Exponents

Suitable Grade Level

Grades 8-10