Simplify 4 cube root of 864x^5y^14

Solution:

Step 1: Decompose the expression

Express $864x^5{y}^{14}$ as a product of cubes and remaining factors.

Step 1.1: Extract the cube factor from 864

Factor out $216$ from $864$: $4\cdot \sqrt[3]{216\cdot 4x^5{y}^{14}}$

Step 1.2: Represent 216 as a cube

Express $216$ as $6^3$: $4\cdot \sqrt[3]{6^3\cdot 4x^5{y}^{14}}$

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: $4\cdot \sqrt[3]{6^3(x^3)(y^4{)}^3\cdot 4x^2{y}^2}$

Step 1.8: Combine into a cube

Combine to form a cube: $4\cdot \sqrt[3]{(6x{y}^4{)}^3\cdot 4x^2{y}^2}$

Step 1.9: Enclose with parentheses

Add parentheses: $4\cdot \sqrt[3]{((6x{y}^4{)}^3\cdot 4(x^2{y}^2))}$

Step 1.10: Finalize the expression

Ensure proper parentheses: $4\cdot \sqrt[3]{((6x{y}^4{)}^3\cdot (4x^2{y}^2))}$

Step 2: Simplify the radical

Extract terms from under the cube root: $4(6x{y}^4)\cdot \sqrt[3]{4x^2{y}^2}$

Step 3: Multiply the coefficients

Combine the coefficients: $24x{y}^4\cdot \sqrt[3]{4x^2{y}^2}$

Knowledge Notes:

To simplify an expression involving cube roots, we can use the following knowledge points:

1. Factorization: Breaking down a number into its prime factors or other suitable factors that can simplify the expression.

2. 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.

3. Cube Roots: Recognizing that $\sqrt[3]{a^3}=a$ allows us to simplify cube roots when the radicand is a perfect cube.

4. Combining Like Terms: When terms share the same variable and exponent, they can be combined by multiplying or adding the coefficients.

5. 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.