Simplify ( square root of 3(|x|-5x^2))/3
The given problem involves expressing a mathematical expression in a simpler form. The expression includes the square root of a rational function, where the numerator consists of the absolute value of a linear term (|x|) subtracted from a quadratic term (-5x^2). This entire numerator is then divided by 3, which is the denominator. The task is to simplify this complex expression to possibly remove the square root, simplify the fraction, or make it easier to interpret or calculate.
$\frac{\sqrt{3 \left(\right. \left|\right. x \left|\right. - 5 x^{2} \left.\right)}}{3}$
Answer
Solution:
Step 1: Simplify the expression
Given the expression $\frac{\sqrt{3(|x| - 5x^2)}}{3}$, we aim to simplify it.
Step 2: Factor out the common term
We notice that both the numerator and the denominator have a common factor of 3. We can simplify the expression by canceling out this common factor.
Step 3: Simplify the square root
After canceling out the common factor, we are left with $\sqrt{|x| - 5x^2}$.
Step 4: Final simplification
Since there is no further simplification possible, the simplified form of the given expression is $\sqrt{|x| - 5x^2}$.
Knowledge Notes:
Absolute Value: The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted by two vertical bars, e.g., $|x|$.
Square Root: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because $3^2 = 9$.
Simplification of Radicals: When simplifying expressions involving square roots, one looks for factors that are perfect squares and simplifies them outside the radical.
Rationalizing the Denominator: If a radical is present in the denominator, the process of eliminating it is called rationalizing the denominator. However, in this problem, we have a square root in the numerator, not the denominator.
Factoring: Factoring involves rewriting an expression as a product of its factors. It can simplify expressions, especially when common factors are present in both the numerator and the denominator, allowing for cancellation.
Cancellation: When the same factor appears in both the numerator and the denominator of a fraction, it can be canceled out, simplifying the fraction. This is based on the property that a number divided by itself equals one.
