Simplify 7 cube root of bc- cube root of bc+10 cube root of bc

Solution:

Step 1:

First, remove $\sqrt[3]{bc}$ from $7\sqrt[3]{bc}$ to obtain $6\sqrt[3]{bc}$, then add this to $10\sqrt[3]{bc}$.

Resulting expression: $6\sqrt[3]{bc} + 10\sqrt[3]{bc}$

Step 2:

Combine the like terms $6\sqrt[3]{bc}$ and $10\sqrt[3]{bc}$ to get their sum.

Final Simplified Form: $16\sqrt[3]{bc}$

Knowledge Notes:

To simplify an expression involving cube roots, you need to understand the following concepts:

1. Cube Root: The cube root of a number $x$, denoted as $\sqrt[3]{x}$, is a value that, when multiplied by itself three times, gives the number $x$. For instance, $\sqrt[3]{27} = 3$ because $3 \times 3 \times 3 = 27$.

2. Like Terms: In algebra, like terms are terms that have the same variable raised to the same power. In this case, all terms are cube roots of $bc$, so they are like terms and can be combined.

3. Combining Like Terms: This is a process used in algebra to simplify expressions by adding or subtracting like terms. For example, $a\sqrt[3]{x} + b\sqrt[3]{x} = (a+b)\sqrt[3]{x}$, where $a$ and $b$ are coefficients.

4. Subtraction and Addition of Like Terms: When subtracting or adding like terms, you only combine the coefficients (the numerical part in front of the cube root) and keep the cube root part unchanged.

In the given problem, the steps involve subtracting and then adding the coefficients of the like terms which are cube roots of $bc$. The subtraction in Step 1 simplifies $7\sqrt[3]{bc} - \sqrt[3]{bc}$ to $6\sqrt[3]{bc}$, and the addition in Step 2 combines $6\sqrt[3]{bc} + 10\sqrt[3]{bc}$ to $16\sqrt[3]{bc}$.