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A L G E B R A

27

Simplest form

2nd level

Simplifying the square roots of powers

WE SAY THAT A SQUARE ROOT RADICAL is simplified, or in its simplest form, when the radicand has no square factors.

A radical is also in simplest form when the radicand is not a fraction.

Example 1.   33, for example, has no square factors.  Its factors are 3 · 11, neither of which is a square number. Therefore, is in its simplest form.

Example 2.  Extracting the square root.   18 has the square factor 9.

18 = 9 · 2.

Therefore, is not in its simplest form.  We have,

=

We may now extract, or take out, the square root of 9:

= = 3.

is now simplified. The radicand no longer has any square factors.

The justification for taking out the square root of 9, is this theorem:

The square root of a product
is equal to the product of the square roots
of each factor.

We will prove that when we come to rational exponents, Lesson 29. Here is a simple illustration:

As for , then, it is equal to the square root of 9  times the square root of 2, which is irrational.  3.

Example 3   Simplify .

Solution.    = = 5.

75 has the square factor 25.  And the square root of (25 times 3)
is equal to the square root of 25  times the square root of 3.

is now simplified.

Example 4.   Simplify .

Solution.   We have to factor 42 and see if it has any square factors.  We can begin the factoring in any way.  For example,

42 = 6 · 7

We can continue to factor 6 as 2 · 3, but we cannot continue to factor 7 because 7 is a prime number. Therefore,

42 = 2 · 3 · 7

We now see that, because no factor is repeated, 42 has no square factors. therefore is in its simplest form.

Compare Example 1 and Problem 2 of the previous Lesson.

· 3 · 7 is the prime factorization of 42.

2, 3, and 7 are prime numbers.

Example 5.   Simplify .

Solution.  We must look for square factors, which will be factors that are repeated.

180 = 2 · 90 = 2 · · 45 = 2 · 2 · · 5 = 2 · 2 · · 3 · 5

Therefore,

= 2 · 3 = 6.

Problem 1.   To simplify a radical, why do we look for square factors?

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Do the problem yourself first!

In order to take its square root out of the radical.

 Problem 2.   Which is correct?

Problem 3.   Simplify the following.  Do that by inspecting each radicand for a square factor:  4, 9, 16, 25, and so on.

a)    =

b)    =  = = 5

c)    =  = = 3

d)    =  = 7

e)     =  = 4

f)    =  = 10

g)    =  = 5

h)    =  = 4

Reduce to lowest terms.

 a) 2 = 2 = 2 =
 b) 3 = 3 = 3 = 2
 c) 2 = The radical is in its simplest form. The fraction cannot be reduced.

Similar radicals have the same radicand.  We add them as like terms.

 7 + 2 + 5 + 6 − = 7 + 2 + 6 + 5 − = 7 + 8 + 4.

2 and 6 are similar, as are 5 and −.  We combine them by adding their coefficients.

In practice, it is not necessary to change the order of the terms. The student should simply see which radicals have the same radicand.

As for 7, it does not "belong" to any radical.

Problem 5.   Simplify each radical, then add the similar radicals.

a)   + = 3 + 2 = 5

 b)   4 − 2 + = 4 − 2 + = 4· 5 − 2· 7 + = 20 − 14 + = 7
 c)   3 + − 2 = 3 + − 2 = 3· 2 + 2 − 2· 4 = 6 + 2 − 8 = 2 − 2
 d)   3 + + = 3 + + = 3 + 2 + 3 = 3 + 5
 e)   1 − + = 1 − + = 1 − 8 + 3 = 1 − 5

Problem 6.   Simplify the following.

 a) 2 = 2 = 2 − , on dividing every term in the numerator by 2.

Compare Example 4 here.

To see that 2 was a factor of the radical, you first have to simplify the radical.  Compare Problem 4.

 b) 5 = 5 = 2 +
 c) 6 = 6 = 3 on dividing every term by 2.

2nd Level

First Lesson on Radicals

Next Lesson:  Multiplying and dividing radicals

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