Math Problem Statement

Simplify the expression: ∛(6x^3 y^7) · ∛(4x^5)

Solution

The expression you provided is:

[ \sqrt[3]{6x^3 y^7} \cdot \sqrt[3]{4x^5} ]

Simplifying the expression:

We will apply the property of cube roots that allows us to combine them under one radical when multiplied:

[ \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b} ]

Thus, we can rewrite the expression as:

(6x3y7)(4x5)3\sqrt[3]{(6x^3 y^7) \cdot (4x^5)}

Multiply inside the radical:

6x3y74x5=(64)(x3x5)y7=24x8y76x^3 y^7 \cdot 4x^5 = (6 \cdot 4) \cdot (x^3 \cdot x^5) \cdot y^7 = 24x^8 y^7

Now we have:

24x8y73\sqrt[3]{24x^8 y^7}

Simplifying the cube root:

Now, let's extract cubes from under the radical:

  1. x8=x32+2=(x6)x2x^8 = x^{3 \cdot 2 + 2} = (x^6) \cdot x^2, so we can take out x2x^2 since x6x^6 is a perfect cube.
  2. y7=y32+1=(y6)yy^7 = y^{3 \cdot 2 + 1} = (y^6) \cdot y, so we can take out y2y^2 since y6y^6 is a perfect cube.

Thus, the expression becomes:

x2y224x2y3x^2 y^2 \sqrt[3]{24 x^2 y}

This is the simplified form of the given expression.

Would you like further details or have any questions?

Related Questions:

  1. How do you simplify expressions with higher roots (e.g., fourth or fifth roots)?
  2. Can you explain how to handle fractional exponents when simplifying radicals?
  3. What other properties of radicals can help in simplifying complex expressions?
  4. How can you use prime factorization to simplify cube roots?
  5. What are the applications of radical simplification in real-world problems?

Tip: When simplifying radicals, always look for perfect squares, cubes, etc., within the expression to make it easier to extract values from the radical.

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Math Problem Analysis

Mathematical Concepts

Cube Roots
Multiplication of Radicals
Exponent Rules
Simplifying Expressions

Formulas

∛a · ∛b = ∛(a · b)
x^m · x^n = x^(m+n)
Extract cubes: x^6 = (x^3)^2

Theorems

Properties of Radicals
Laws of Exponents

Suitable Grade Level

Grades 9-12