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------------------------------------------------ What is the simplest radical form of the expression [tex]23(x^5y^3)^{\frac{2}{3}}[/tex]?

A) [tex]x^2y\sqrt[3]{y^2}[/tex]

B) [tex]x^4y^4\sqrt{y}[/tex]

C) [tex]x^3y^2\sqrt[3]{x}[/tex]

D) [tex]x^7y^4\sqrt{xy}[/tex]

The simplest radical form of the expression 23(x⁵y³)²／³ is x³³⁄₃y². None of the choices provided are correct. This answer was found by using the rules of exponents to first square the expression in the parentheses and then take the cube root.

Explanation:

To simplify the expression 23(x⁵y³)²／³, we first square the expression inside the parentheses. By the rule of exponents, (x^a)^b = x^(a*b), we have (x⁵y³)² = x¹⁰y⁶. Then we take the cube root of this expression, again using the rule of exponents, we have (x¹⁰y⁶)⅓ = x¹⁰/₃y⁶/₃ = x³³⁄₃y². So the simplest radical form of the expression is x³³⁄₃y². None of the choices given (A, B, C, D) are correct.

The simplest radical form of the expression 23(x⁵y³)²／³ is A) x²y∛ y².

To simplify the expression, we can use the property of exponents that states (a^m)^n = a^(m*n). Applying this property, we have:

23(x⁵y³)²／³ = 23(x^(5*2) * (y³)^(2/3)) = 23(x^10 * y^(6/3)) = 23(x^10 * y^2).

Therefore, the simplest radical form is A) x²y∛ y².

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