What Is the Simplified Form of the Following Expression?
In mathematics, simplification is the process of reducing a mathematical expression to its most basic form. This can be done by combining like terms, factoring, or using other algebraic techniques.
One common type of simplification problem involves expressions with radicals. A radical is a symbol that represents the square root, cube root, or other root of a number. For example, the expression $\sqrt{16}$ is simplified to 4, because 4 is the square root of 16.
To simplify an expression with radicals, we can use the following steps:
- Factor the expression inside the radical. This will often result in a simpler expression inside the radical, which can then be simplified further.
- Combine like terms. This can be done with both the terms inside and outside the radical.
- Use the properties of radicals. For example, $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.
Here are some examples of how to simplify expressions with radicals:
Example 1:
$\sqrt{12x^2}$ Factor 12 to get 2 * 2 * 3.
$\sqrt{12x^2} = \sqrt{2^2 \cdot 3 \cdot x^2}$ Use the property that $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.
$\sqrt{12x^2} = \sqrt{2^2} \cdot \sqrt{3} \cdot \sqrt{x^2}$ Simplify the terms inside the radical.
$\sqrt{12x^2} = 2 \sqrt{3} \cdot x$ The simplified form of the expression is $2 \sqrt{3}x$.
Example 2:
$\sqrt{4x^2 + 9}$ Factor 4 and 9 to get 2 * 2 and 3 * 3.
$\sqrt{4x^2 + 9} = \sqrt{(2^2)x^2 + (3^2)}$ Use the property that $\sqrt{a^2 + b^2} = a + b$.
$\sqrt{4x^2 + 9} = \sqrt{x^2 + 3^2} = x + 3$ The simplified form of the expression is $x + 3$.
Here are some questions that you might be asked about simplifying expressions with radicals:
- What is the simplified form of $\sqrt{36x^4}$?
- What is the simplified form of $\sqrt{16x^2 + 81}$?
- What is the simplified form of $\sqrt{2x^2 + 3x + 9}$?
To answer these questions, you can use the steps outlined above.
