Math Problem Statement

Write \((4y^4)^2\) without exponents.

Solution

To solve (4y4)2\left(4y^4\right)^2 without using exponents, we can apply the power rule for exponents, which states that (am)n=amn(a^m)^n = a^{m \cdot n}.

  1. Expand the expression using the rule: (4y4)2=42(y4)2\left(4y^4\right)^2 = 4^2 \cdot (y^4)^2

  2. Calculate 424^2: 42=164^2 = 16

  3. Apply the power rule to (y4)2(y^4)^2: (y4)2=y42=y8(y^4)^2 = y^{4 \cdot 2} = y^8

  4. Combine the results: (4y4)2=16y8\left(4y^4\right)^2 = 16y^8

Answer: The expression without exponents is 16y816y^8.

Would you like further details, or do you have additional questions?


Here are some related questions to expand on the topic:

  1. How does the power rule for exponents work with negative exponents?
  2. What is the general form of the power rule when there are multiple variables?
  3. How would you simplify (3x2y3)4\left(3x^2y^3\right)^4?
  4. Can the power rule be applied to fractional exponents?
  5. What happens when the exponent is zero in expressions like (am)0(a^m)^0?

Tip: When applying the power rule, remember to multiply the exponents to keep the expression simplified.

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Math Problem Analysis

Mathematical Concepts

Exponents
Power Rules
Algebra

Formulas

(a^m)^n = a^{m \cdot n}

Theorems

Power Rule for Exponents

Suitable Grade Level

Grades 8-10