Math Problem Statement

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(4th root of ab)^2

Solution

The given mathematical expression is:

(ab4)2\left( \sqrt[4]{ab} \right)^2

Step-by-Step Explanation:

  1. Understand the Expression:

    • The term ab4\sqrt[4]{ab} represents the fourth root of the product abab.
    • Squaring the fourth root means raising it to the power of 22.
  2. Simplify the Powers:

    • Using the rule of exponents (xm)n=xmn(x^{m})^n = x^{m \cdot n}: (ab4)2=((ab)1/4)2=(ab)(1/4)2.\left( \sqrt[4]{ab} \right)^2 = \left( (ab)^{1/4} \right)^2 = (ab)^{(1/4) \cdot 2}.
  3. Simplify the Exponent:

    • Multiply the exponents: (ab)(1/4)2=(ab)1/2.(ab)^{(1/4) \cdot 2} = (ab)^{1/2}.
  4. Result:

    • The expression (ab)1/2(ab)^{1/2} is equivalent to: ab.\sqrt{ab}.

Final Answer:

(ab4)2=ab.\left( \sqrt[4]{ab} \right)^2 = \sqrt{ab}.

Would you like further clarification or a breakdown of the rules used? Here are five related questions for practice:

  1. How do you simplify (a23)3(\sqrt[3]{a^2})^3?
  2. What is (ab2)1/2(ab^2)^{1/2} in radical form?
  3. Explain the general rule for simplifying (x1/n)m(x^{1/n})^m.
  4. How does (a1/3b1/4)2(a^{1/3}b^{1/4})^2 simplify?
  5. Why does (a4)4=a(\sqrt[4]{a})^4 = a?

Tip: Always remember that roots can be expressed as fractional exponents, which simplifies operations like multiplication or division.

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Math Problem Analysis

Mathematical Concepts

Roots and Exponents
Simplification of Radical Expressions

Formulas

Exponentiation rule: (x^m)^n = x^(m*n)
Roots as fractional exponents: nth root of x = x^(1/n)

Theorems

Properties of Exponents

Suitable Grade Level

Grades 9-10