Simplify cube root of 64(x+y)^6

The given problem is a mathematical expression simplification task. It involves calculating the cube root of a quantity that contains both a definite number (64) and a variable expression raised to a power ((x+y)^6). The requester is asking to perform the operations necessary to simplify the expression using algebraic rules and the properties of exponents and roots.

$\sqrt[3]{64 {((x + y))}^{6}}$

Answer

Expert–verified

Solution:

Step 1:

Express $64 (x + y)^{6}$ as $(4 (x + y)^{2})^{3}$ and take the cube root: $\sqrt[3]{(4 (x + y)^{2})^{3}}$ .

Step 2:

Extract terms from under the cube root, assuming all numbers are real: $4 (x + y)^{2}$ .

Step 3:

Represent $(x + y)^{2}$ as the product of $(x + y)$ with itself: $4 ((x + y) (x + y))$ .

Step 4:

Expand $(x + y) (x + y)$ by using the FOIL Method.

Step 4.1:

Use the distributive property: $4 (x (x + y) + y (x + y))$ .

Step 4.2:

Distribute $x$ over $(x + y)$ : $4 (x x + x y + y (x + y))$ .

Step 4.3:

Distribute $y$ over $(x + y)$ : $4 (x x + x y + y x + y y)$ .

Step 5:

Combine like terms and simplify.

Step 5.1:

Simplify each term individually.

Step 5.1.1:

Multiply $x$ by $x$ : $4 (x^{2} + x y + y x + y y)$ .

Step 5.1.2:

Multiply $y$ by $y$ : $4 (x^{2} + x y + y x + y^{2})$ .

Step 5.2:

Combine the terms $x y$ and $y x$ .

Step 5.2.1:

Rearrange $y x$ to $x y$ : $4 (x^{2} + x y + x y + y^{2})$ .

Step 5.2.2:

Combine $x y$ and $x y$ : $4 (x^{2} + 2 x y + y^{2})$ .

Step 6:

Apply the distributive property: $4 x^{2} + 4 (2 x y) + 4 y^{2}$ .

Step 7:

Multiply $2$ by $4$ : $4 x^{2} + 8 x y + 4 y^{2}$ .

Knowledge Notes:

To simplify the cube root of a number or expression raised to a power that is a multiple of 3, we can rewrite the expression in a form that makes it easier to apply the cube root. For example, $64 (x + y)^{6}$ can be rewritten as $(4 (x + y)^{2})^{3}$ because $64$ is $4^{3}$ and $(x + y)^{6}$ is $((x + y)^{2})^{3}$ . This allows us to directly apply the cube root to eliminate the exponent of 3, simplifying the expression.

When expanding a binomial squared, such as $(x + y)^{2}$ , we can use the FOIL (First, Outer, Inner, Last) method to multiply each term in the first binomial by each term in the second binomial. The distributive property is used here to ensure all terms are multiplied correctly.

Combining like terms is a crucial step in simplification. Terms with the same variables raised to the same power can be combined by adding their coefficients. For instance, $x y$ and $y x$ are like terms because they represent the same product of variables, regardless of the order.

Finally, the distributive property is used to multiply a constant outside the parentheses by each term inside the parentheses. This step is essential in expanding expressions and simplifying them further.