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How to Simplify $4x^{2}\\sqrt[3]{27x^{5}y^{9}}$ to Simplest Radical Form
Mathematics
Grade 10 of Junior High School
Question Content
Express the following in simplest radical form: $4x^{2}\\sqrt[3]{27x^{5}y^{9}}$
Correct Answer
$12x^{3}y^{3}\\sqrt[3]{x^{2}}$
Detailed Solution Steps
1
Step 1: Simplify the cube root term by breaking down the radicand into perfect cube factors and remaining factors. First, factor each part: $27 = 3^3$, $x^5 = x^3 \\cdot x^2$, $y^9 = (y^3)^3$. So $\\sqrt[3]{27x^{5}y^{9}} = \\sqrt[3]{3^3 \\cdot x^3 \\cdot x^2 \\cdot (y^3)^3}$.
2
Step 2: Use the property of radicals $\\sqrt[n]{ab} = \\sqrt[n]{a} \\cdot \\sqrt[n]{b}$ to separate the perfect cube factors: $\\sqrt[3]{3^3} \\cdot \\sqrt[3]{x^3} \\cdot \\sqrt[3]{(y^3)^3} \\cdot \\sqrt[3]{x^2}$.
3
Step 3: Simplify the perfect cube roots: $\\sqrt[3]{3^3}=3$, $\\sqrt[3]{x^3}=x$, $\\sqrt[3]{(y^3)^3}=y^3$. This simplifies the radical term to $3xy^3\\sqrt[3]{x^2}$.
4
Step 4: Multiply the simplified radical term with the coefficient outside the radical: $4x^2 \\times 3xy^3\\sqrt[3]{x^2}$. Multiply the coefficients and combine like variables using exponent rules ($x^a \\times x^b = x^{a+b}$): $12x^{2+1}y^3\\sqrt[3]{x^2} = 12x^3y^3\\sqrt[3]{x^2}$.
Knowledge Points Involved
1
Perfect Cube Factors
A perfect cube is a number or expression that can be written as $n^3$ where $n$ is an integer or monomial. For radical simplification, we break radicands into perfect cube factors to pull them out of the radical. For example, $27=3^3$, $y^9=(y^3)^3$ are perfect cubes.
2
Properties of Cube Roots
The property $\\sqrt[3]{ab} = \\sqrt[3]{a} \\cdot \\sqrt[3]{b}$ allows us to split a cube root of a product into the product of cube roots. Also, $\\sqrt[3]{a^3}=a$ for any real number $a$, which lets us simplify perfect cube terms inside cube roots.
3
Exponent Rules for Multiplying Monomials
When multiplying monomials with the same base, we add their exponents: $x^a \\times x^b = x^{a+b}$. This is used to combine like variable terms after simplifying the radical.
4
Simplest Radical Form for Cube Roots
A cube root is in simplest form when there are no perfect cube factors left inside the radical, no fractions inside the radical, and no radicals in the denominator. For this problem, we leave $\\sqrt[3]{x^2}$ since $x^2$ has no perfect cube factors.
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