Simplify 4 cube root of 864x^5y^14
The question is asking for the simplification of a mathematical expression. Specifically, it involves finding the cube root of the numerical value 864 and the variables x^5 and y^14, and then multiplying the result by 4. The goal is to express the original expression in a simpler form by performing the cube root operation and simplifying any resulting coefficients or exponents.
$4 \sqrt[3]{864 x^{5} y^{14}}$
Answer
Solution:
Step 1: Decompose the expression
Express $864 x^{5} y^{14}$ as a product of cubes and remaining factors.
Step 1.1: Extract the cube factor from 864
Factor out $216$ from $864$: $4 \cdot \sqrt[3]{216 \cdot 4 x^{5} y^{14}}$
Step 1.2: Represent 216 as a cube
Express $216$ as $6^3$: $4 \cdot \sqrt[3]{6^3 \cdot 4 x^{5} y^{14}}$
Step 1.3: Separate $x^3$ from $x^5$
Factor $x^3$ from $x^5$: $4 \cdot \sqrt[3]{6^3 \cdot 4 (x^3 x^2) y^{14}}$
Step 1.4: Separate $y^{12}$ from $y^{14}$
Factor $y^{12}$ from $y^{14}$: $4 \cdot \sqrt[3]{6^3 \cdot 4 (x^3 x^2) (y^{12} y^2)}$
Step 1.5: Express $y^{12}$ as a cube
Rewrite $y^{12}$ as $(y^4)^3$: $4 \cdot \sqrt[3]{6^3 \cdot 4 (x^3 x^2) ((y^4)^3 y^2)}$
Step 1.6: Rearrange $x^2$
Move $x^2$: $4 \cdot \sqrt[3]{6^3 \cdot 4 (x^3) (y^4)^3 x^2 y^2}$
Step 1.7: Rearrange the factor 4
Move the factor 4: $4 \cdot \sqrt[3]{6^3 (x^3) (y^4)^3 \cdot 4 x^2 y^2}$
Step 1.8: Combine into a cube
Combine to form a cube: $4 \cdot \sqrt[3]{(6 x y^4)^3 \cdot 4 x^2 y^2}$
Step 1.9: Enclose with parentheses
Add parentheses: $4 \cdot \sqrt[3]{((6 x y^4)^3 \cdot 4 (x^2 y^2))}$
Step 1.10: Finalize the expression
Ensure proper parentheses: $4 \cdot \sqrt[3]{((6 x y^4)^3 \cdot (4 x^2 y^2))}$
Step 2: Simplify the radical
Extract terms from under the cube root: $4 (6 x y^4) \cdot \sqrt[3]{4 x^2 y^2}$
Step 3: Multiply the coefficients
Combine the coefficients: $24 x y^4 \cdot \sqrt[3]{4 x^2 y^2}$
Knowledge Notes:
To simplify an expression involving cube roots, we can use the following knowledge points:
Factorization: Breaking down a number into its prime factors or other suitable factors that can simplify the expression.
Properties of Exponents: Understanding that $a^{m \cdot n} = (a^m)^n$ and $a^{m+n} = a^m \cdot a^n$ helps to manipulate and simplify expressions with exponents.
Cube Roots: Recognizing that $\sqrt[3]{a^3} = a$ allows us to simplify cube roots when the radicand is a perfect cube.
Combining Like Terms: When terms share the same variable and exponent, they can be combined by multiplying or adding the coefficients.
Algebraic Manipulation: Rearranging terms and factors to simplify the expression and make extraction of cubes more apparent.
By applying these principles, we can simplify complex expressions involving cube roots and exponents.
