Download Multiplication Property of Square Roots For all number a and b

Simplifying Radicals Packet
Simplest Radical Form
Radical expressions that are simplified are easier to manipulate algebraically. A square-root expression is in
simplest radical form when all of the following conditions are met:



No factor of the radicand is a perfect square other than 1.
The radicand contains no fractions.
No radical appears in the denominator of a fraction.
Multiplication Property of Square Roots
For all number a and b, where a≥ 0 and b≥ 0: √𝑎𝑏 = √𝑎√𝑏
√12 To simplify this radical, look for perfect-square factors, and apply the Multiplication Property of
Square Roots. Then find the square roots of the perfect squares. Leave any factor that is not a perfect
square in radical form.
√12 = √4 ∙ 3 = √4 ∙ √3 = 2√3
√18 = √9 ∙ 2 = √9 ∙ √2 = 3√2
In these examples it’s easy to identify the perfect-square within the number, but as the numbers get larger,
you need to find a way to find the perfect-square factors. A factor ladder can help! A factor ladder is similar
to prime factorization except each time you factor out the lowest prime number.
80
√80 = √22 ∙ 22 ∙ 5 = √22 ∙ √22 ∙ √5 = 2 ∙ 2√5 = 4√5
75
Another condition that must be met to be in simplest radical form is that there is no radical in the
denominator of a fraction. Since an irrational number can’t be in the denominator, we call this process
rationalizing the denominator.
Rationalizing the denominator
3
5
7
5
√2
3
Multiplying/Dividing Radicals
You’ll use the multiplication property of square roots to simplify radicals.
8  14  8 14  112
Are we done? No! We have to make sure the product is also in simplest radical form.
√112 = √22 ∙ 22 ∙ 7 = 2 ∙ 2√7 = 4√7
3 6
3 5 
2
Division Property of Square Roots
𝑎
For all numbers 𝑎 ≥ 0 and 𝑏 > 0: √ =
𝑏
√
6
49
=
√6
√49
=
√6
7
√𝑎
√𝑏
√
225
18
Remember, simplest form means nothing else can be done! Watch out for common factors in the
coefficients!
Radical Operations 2
In order to add or subtract expressions that contain radicals, radicands must be equal (like terms). Terms
that contain√3, for example, can be combined by applying the Distributive Property.
7 3  9 3  (7  9) 3  2 3
You might not be able to see if the radicals are equal until they are in simplest form. First put both in SRF.
√72 + √32
For this type of problem, you’ll multiply with the distributive property, factor, and simplify the result.
√2(6 + √12)
2
 21 3 14 
Simplifying Radicals 3
You can also simplify radicals that contain constants and variables. You’ll use the same methods we’ve
learned for the constants. For the variables we’ll use a modification of the rules of exponents.
49a 6b 4  (7) 2 (a3 ) 2 (b 2 ) 2  7a3b 2
Notice how you can use the “opposite” of the power to a power property to identify the perfect squares
under the radical.
√81𝑥 4 𝑦10
Where you put the parentheses is important to this method. Put the perfect squares in the parentheses.
You’ll always have an exponent of 2 on the outside of the parentheses!
Using this method makes it easy to identify what you are taking out of the radical and what is staying in the
radical.
Sometimes you’ll have “leftovers” after you’ve identified the perfect squares. The leftovers stay in the
radical.
√225𝑎5 𝑏 7
And sometimes the constant will not be a perfect square so you’ll have to simplify that as well!
√32𝑥 6 𝑦 3
And finally, all of these work for cube roots also! This time you’ll be looking for perfect cubes under the
radical.
3
3
√8𝑎9 𝑏15 = √(2)3 (𝑎3 )3 (𝑏 5 )3 = 2𝑎3 𝑏 3
3
√54𝑎14 𝑏18