Radicals cannot appear in the denominator of a fraction.
To write a radical expression in the simplest form, we need to check each point from the list. For every condition that is not fulfilled, we should take appropriate steps to fix it. We will consider each point and present examples to illustrate the methods available to fix these issues.
Perfect Square Factors
When simplifying a radical, we begin by checking if we can rewrite the numbers or variables under the root as the product of at least one perfect square factor. If we can, we use the Product Property of Square Roots to split the radical into the product of radicals. This way we can simplify the expression by calculating the square root of the radicand which is a perfect square.
Let's look at a few more examples. Remember that when finding the principal square root of an expression containing variables, we have to be sure that the result is not negative. Therefore, sometimes we need to use the absolute value symbol.
Before Simplification
Product of Radicals
After Simplification
sqrt(20)
sqrt(4) * sqrt(5)
2sqrt(5)
sqrt(81y)
sqrt(81) * sqrt(y)
9sqrt(y)
sqrt(2x^2)
sqrt(x^2) * sqrt(2)
|x|sqrt(2)
sqrt(a^3)
sqrt(a^2) * sqrt(a)
|a|sqrt(a)
Fractions Inside Radicals
Next, we need to check if there are any fractions inside the radicals. If there are, we should use the Quotient Property of Square Roots to rewrite the radical as the division of two radicals.
2sqrt(2x/3)=2sqrt(2x)/sqrt(3)
Let's look at a couple more examples.
Before Simplification
After Simplification
sqrt(5/7)
sqrt(5)/sqrt(7)
sqrt(8x/3y)
sqrt(8x)/sqrt(3y)
Radicals in the Denominator
Finally, if there are any radicals in the denominator, we need to rationalize it.
Monomial in the Denominator
For a monomial denominator, we expand the fraction by a radical that will eliminate the radical in the denominator. When there is a square root in the denominator, we need to expand the fraction by a square root that will give us a perfect square in the radicand.
2sqrt(2x)/sqrt(3) * sqrt(3)/sqrt(3) = 2sqrt(2x * 3)/sqrt(3 * 3) = 2sqrt(6x)/3
Let's look at a few more examples. Again we need to be careful when calculating the square root of an expression containing a variable because the result cannot be negative.
Before Simplification
Multiplication
After Simplification
sqrt(3)/sqrt(11)
sqrt(3)/sqrt(11) * sqrt(11)/sqrt(11)
sqrt(33)/11
sqrt(5)/sqrt(6)
sqrt(5)/sqrt(6) * sqrt(6)/sqrt(6)
sqrt(30)/6
sqrt(7x)/sqrt(3)
sqrt(7x)/sqrt(3) * sqrt(3)/sqrt(3)
sqrt(21x)/3
sqrt(2)/sqrt(x)
sqrt(2)/sqrt(x) * sqrt(x)/sqrt(x)
sqrt(2x)/|x|
Binomial in the Denominator
When there is a binomial denominator, we multiply the numerator and denominator of the fraction by the conjugate of the denominator. We find the conjugate by changing the sign of the second term of the expression.
