Simplify the Radical Expression 6 square root of 10+5 square root of 10-15 square root of 10

Solution:

Step 1:

Combine the like terms $6 \sqrt{10}$ and $5 \sqrt{10}$ to get $11 \sqrt{10}$. Then subtract $15 \sqrt{10}$ from the result.

Step 2:

Perform the subtraction to find the difference between $11 \sqrt{10}$ and $15 \sqrt{10}$, which gives $- 4 \sqrt{10}$.

Step 3:

Express the final answer in its exact form and decimal approximation.

Exact Form: $- 4 \sqrt{10}$ Decimal Form: Approximately $- 12.64911064$

Knowledge Notes:

1. Like terms in algebra are terms that have the same variables and powers. The coefficients do not need to be the same. In this problem, $6 \sqrt{10}$, $5 \sqrt{10}$, and $-15 \sqrt{10}$ are like terms because they all contain the radical $\sqrt{10}$.

2. When adding or subtracting like terms, you combine or subtract their coefficients while keeping the variable part the same. For example, $a \sqrt{b} + c \sqrt{b} = (a+c) \sqrt{b}$.

3. The square root function, denoted by $\sqrt{\phantom{0}}$, is a radical function that finds a number which, when multiplied by itself, gives the original number under the radical sign. For instance, $\sqrt{10}$ is the number which, when squared, equals 10.

4. To simplify radical expressions, you combine like terms and perform arithmetic operations as you would with regular numbers, keeping the radical part intact unless it simplifies further (e.g., $\sqrt{9}$ simplifies to 3 because 3 is the square root of 9).

5. The exact form of a radical expression is the simplest form in terms of radicals without decimal approximation. The decimal form is the numerical approximation of the radical expression, often rounded to a certain number of decimal places.