Math Problem Statement
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:
Multiply inside the radical:
Now we have:
Simplifying the cube root:
Now, let's extract cubes from under the radical:
- , so we can take out since is a perfect cube.
- , so we can take out since is a perfect cube.
Thus, the expression becomes:
This is the simplified form of the given expression.
Would you like further details or have any questions?
Related Questions:
- How do you simplify expressions with higher roots (e.g., fourth or fifth roots)?
- Can you explain how to handle fractional exponents when simplifying radicals?
- What other properties of radicals can help in simplifying complex expressions?
- How can you use prime factorization to simplify cube roots?
- 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