Simplifying radicals, also known as simplifying radical expressions, is the process of rewriting a radical into its most basic form. This involves ensuring that the number under the square root symbol, called the radicand, does not have any perfect square factors. By extracting these perfect square factors from under the root, we reduce the number to its smallest possible form.
The radical symbol, or square root symbol, is represented as $sqrt{;;;}$.
To effectively simplify radicals, we can utilize three key properties derived from the laws of exponents:
Related reading: Laws of Exponents
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Product Property of Radicals: $sqrt{m} times sqrt{n}=sqrt{m n}$
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Quotient Property of Radicals: $sqrt{m} div sqrt{n}=sqrt{cfrac{m}{n}}=cfrac{sqrt{m}}{sqrt{n}}$
In this division, $sqrt{m}$ is the numerator, and $sqrt{n}$ is the denominator.
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Square of a Radical Property: $sqrt{m} times sqrt{m}=sqrt{m^2}=m$
Let’s walk through an example of how to simplify a numerical radical like $sqrt{20}$.
First, we identify that 20 is not a perfect square. To simplify it, we look for perfect square factors of 20.
We can see that 4 is a factor of 20, and importantly, 4 is a perfect square because $2 times 2 =4$.
Therefore, $sqrt{20}$ can be expressed as $sqrt{4 times 5}$.
Using the product property of radicals, we can rewrite this as:
$sqrt{4 times 5}=sqrt{4} times sqrt{5}$
Since $sqrt{4}=2$, we can simplify $sqrt{20}$ to $2 sqrt{5}$.
$sqrt{20}=2 sqrt{5}$
Now, let’s tackle simplifying a radical algebraic expression, such as $sqrt{50 x^3}$.
We start by finding perfect square factors of 50.
25 is a factor of 50, and it is a perfect square number ($5 times 5 = 25$).
Next, we examine the exponent of the variable. An exponent divisible by 2 indicates a perfect square. In $x^3$, the exponent 3 is not divisible by 2, so $x^3$ is not a perfect square.
To handle this, we rewrite $x^3$ as a product where one factor is a perfect square. In this case, $x^3$ can be rewritten as $x^2 times x$.
Now, we can simplify the entire expression step-by-step:
$begin{aligned}&begin{aligned}& sqrt{50 x^3}=sqrt{25 times 2 times x^2 times x} \\ & sqrt{25} times sqrt{2} times sqrt{x^2} times sqrt{x} \\ & sqrt{25}=5end{aligned}\\ &sqrt{x^2}=xend{aligned}$
Thus, $sqrt{50 x^3}$ simplifies to $5 x sqrt{2 x}$.
$sqrt{50 x^3}=5 x sqrt{2 x}$
