Simplify square root of 4x-4
The problem is asking for the simplification of an algebraic expression that contains a square root. Specifically, the expression inside the square root is "4x-4," where "x" represents a variable. The task is to apply the properties of square roots and any relevant algebraic principles to represent the expression in a simpler or more fundamental form, if possible. Simplification might involve factoring, reducing, or otherwise manipulating the expression to achieve a more concise or straightforward representation.
$\sqrt{4 x - 4}$
Answer
Solution:
Simplification Process:
Step 1: Extract the common factor from the terms under the square root.
- Step 1.1: Identify the common factor in $4x$. $\sqrt{4(x - 0)}$
- Step 1.2: Identify the common factor in $-4$. $\sqrt{4(x) + 4(-1)}$
- Step 1.3: Combine the terms with the common factor extracted. $\sqrt{4(x - 1)}$
Step 2: Express the number $4$ as the square of $2$.
- $\sqrt{(2^2)(x - 1)}$
Step 3: Simplify by taking the square root of the perfect square outside the radical.
- $2\sqrt{x - 1}$
Knowledge Notes:
To simplify a square root expression with a factor that is a perfect square, you can use the following steps:
Factorization: Break down the expression under the square root into factors, particularly looking for a perfect square factor.
Square Root of a Perfect Square: Remember that the square root of a perfect square, such as $a^2$, is simply $a$. This is because $\sqrt{a^2} = a$.
Simplification: Once you have a perfect square factor, you can take the square root of that factor outside of the radical sign, simplifying the expression.
Combining Like Terms: If there are like terms under the radical, they can be combined before or after factoring, which can sometimes make the simplification process easier.
In this specific problem, the expression under the square root is $4x - 4$. The number $4$ is a common factor and also a perfect square. By factoring out the $4$, we can then rewrite it as $(2^2)$, which allows us to take the square root of $2^2$ outside the radical, leaving us with $2\sqrt{x - 1}$ as the simplified expression.
