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How to Simplify $4x^2\\sqrt[3]{27x^5y^9}$ to Simplest Radical Form
Mathematics
Grade 10 of Junior High School
Question Content
Express the following in simplest radical form: $4x^2\\sqrt[3]{27x^5y^9}$
Correct Answer
$12x^3y^3\\sqrt[3]{x^2}$
Detailed Solution Steps
1
Step 1: Simplify the cube root term first. Break down the radicand (the expression inside the cube root) into perfect cubes and remaining factors: $\\sqrt[3]{27x^5y^9} = \\sqrt[3]{27 \\times x^3 \\times x^2 \\times y^9}$
2
Step 2: Separate the perfect cube factors from the remaining factors using the property of radicals $\\sqrt[3]{ab} = \\sqrt[3]{a} \\times \\sqrt[3]{b}$: $\\sqrt[3]{27} \\times \\sqrt[3]{x^3} \\times \\sqrt[3]{y^9} \\times \\sqrt[3]{x^2}$
3
Step 3: Calculate the cube root of each perfect cube. $\\sqrt[3]{27}=3$, $\\sqrt[3]{x^3}=x$, $\\sqrt[3]{y^9}=y^3$ (since $(y^3)^3 = y^9$). This simplifies the radical to $3xy^3\\sqrt[3]{x^2}$
4
Step 4: Multiply this simplified radical by the outside term $4x^2$: $4x^2 \\times 3xy^3\\sqrt[3]{x^2}$
5
Step 5: Multiply the coefficients and combine like variable terms using the exponent rule $x^a \\times x^b = x^{a+b}$: $(4\\times3) \\times (x^2 \\times x) \\times y^3\\sqrt[3]{x^2} = 12x^3y^3\\sqrt[3]{x^2}$
Knowledge Points Involved
1
Simplification of Cube Roots
A cube root $\\sqrt[3]{a}$ simplifies to a term where all perfect cube factors are removed from the radical. A perfect cube is a number or expression that can be written as $n^3$ for some integer or monomial $n$. The property $\\sqrt[3]{ab} = \\sqrt[3]{a} \\times \\sqrt[3]{b}$ allows separating factors to isolate perfect cubes, which can then be evaluated as integers or monomials outside the radical.
2
Product Rule for Exponents
The product rule states that when multiplying two terms with the same base, you add their exponents: $x^a \\times x^b = x^{a+b}$. This is used to combine like variable terms when simplifying algebraic expressions, such as multiplying $x^2$ and $x$ to get $x^3$.
3
Monomial Multiplication
When multiplying monomials (terms with a single product of constants and variables), multiply the constant coefficients first, then combine like variable terms using exponent rules. For example, $4x^2 \\times 3xy^3$ is calculated by multiplying 4 and 3, then combining $x^2$ and $x$.
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