Math Problem Statement

(x^2√y)(√[3]{x^3 y^2 z}) / (x^5 y^3)^(1/3)

Solution

The image you uploaded shows a mathematical expression involving exponents and roots.

It appears to be:

(x2y)(x3y2z3)÷(x5/3y3/2)\left( x^2 \sqrt{y} \right) \left( \sqrt[3]{x^3 y^2 z} \right) \div \left( x^{5/3} y^{3/2} \right)

To simplify this expression, let's break it down step by step:

Step 1: Simplify the exponents inside the square and cube roots.

  1. y=y1/2\sqrt{y} = y^{1/2}
  2. x3y2z3=x3/3y2/3z1/3=xy2/3z1/3\sqrt[3]{x^3 y^2 z} = x^{3/3} y^{2/3} z^{1/3} = x y^{2/3} z^{1/3}

Step 2: Combine all terms in the numerator.

The numerator becomes: x2y1/2xy2/3z1/3x^2 \cdot y^{1/2} \cdot x \cdot y^{2/3} \cdot z^{1/3} Which simplifies to: x2+1y1/2+2/3z1/3=x3y7/6z1/3x^{2+1} y^{1/2 + 2/3} z^{1/3} = x^3 y^{7/6} z^{1/3}

Step 3: Combine with the denominator.

Now, the denominator is: x5/3y3/2x^{5/3} y^{3/2} So, the full expression is: x3y7/6z1/3x5/3y3/2\frac{x^3 y^{7/6} z^{1/3}}{x^{5/3} y^{3/2}}

Step 4: Simplify the expression by subtracting exponents for like terms.

For xx: x3÷x5/3=x35/3=x9/35/3=x4/3x^3 \div x^{5/3} = x^{3 - 5/3} = x^{9/3 - 5/3} = x^{4/3}

For yy: y7/6÷y3/2=y7/63/2=y7/69/6=y2/6=y1/3y^{7/6} \div y^{3/2} = y^{7/6 - 3/2} = y^{7/6 - 9/6} = y^{-2/6} = y^{-1/3}

For zz, since there's no zz in the denominator: z1/3z^{1/3}

Final Simplified Expression:

The simplified form of the expression is: x4/3y1/3z1/3x^{4/3} y^{-1/3} z^{1/3}

Would you like to go over the steps in more detail or need further clarification?

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Math Problem Analysis

Mathematical Concepts

Exponentiation
Radicals
Algebraic Simplification

Formulas

√a = a^(1/2)
∛a = a^(1/3)
a^m * a^n = a^(m+n)
a^m / a^n = a^(m-n)

Theorems

Laws of Exponents

Suitable Grade Level

Grades 9-12