# MATHS9 - 2.3 (Surds)

```Exercise 2.3
Surds
Chapter Sections
Introduction to Surds
Simplifying Surds
Multiplying and Dividing Surds
Martin-Gay, Developmental Mathematics
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Introduction to Surds
Square Roots
Opposite of squaring a number is taking the
square root of a number.
A number b is a square root of a number a if b2
= a.
In order to find a square root of a, you need a
# that, when squared, equals a.
Martin-Gay, Developmental Mathematics
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Square Roots
The principal (positive) square root is noted
as
a
The negative square root is noted as
− a
Martin-Gay, Developmental Mathematics
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Square Roots
Example 1: Simplify the following.
Martin-Gay, Developmental Mathematics
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Perfect Squares
Square roots of perfect squares simplify to
rational numbers (numbers that can be written
as a quotient of integers).
Square roots of numbers that are not perfect
squares (like 7, 10, etc.) are irrational
numbers.
IF REQUESTED, you can find a decimal
approximation for these irrational numbers.
Otherwise, leave them in exact form.
Martin-Gay, Developmental Mathematics
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Cube Roots
The cube root of a real number a
3
a = b only if b 3 = a
Note: a is not restricted to non-negative
numbers for cubes.
Martin-Gay, Developmental Mathematics
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Cube Roots
Example 2: Simplify the following.
Martin-Gay, Developmental Mathematics
!9
Simplifying Surds
Product Rule for Surds
If
a and
b are real numbers,
ab = a ⋅ b
a
a
=
if
b
b
b ≠0
Martin-Gay, Developmental Mathematics
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Quotient Rule for Surds
If
n
a and n b are real numbers,
n
n
ab = n a ⋅ n b
a na n
= n if b ≠ 0
b
b
Martin-Gay, Developmental Mathematics
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Simplifying Surds
Example 3: Simplify the following.
Martin-Gay, Developmental Mathematics
!13
Simplifying Surds
Example 4: Simplify the following.
Martin-Gay, Developmental Mathematics
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Sums and Differences
Rules in the previous section allowed us to
split surds which was a product or a quotient.
We can NOT split sums or differences.
a+b ≠ a + b
a −b ≠ a − b
Martin-Gay, Developmental Mathematics
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Like Surds
In previous chapters, we’ve discussed the concept of “like”
terms.
These are terms with the same variables raised to the same
powers.
They can be combined through addition and subtraction.
Similarly, we can work with the concept of “like” surds to
combine surds with the same radicand.
Martin-Gay, Developmental Mathematics
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Example
3+7 3 = 8 3
10 2 − 4 2 = 6 2
5+ 3
Can not simplify
Martin-Gay, Developmental Mathematics
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Example 5: Simplify the following.
Martin-Gay, Developmental Mathematics
!19
Multiplying and Dividing Surds
Multiplying and Dividing Surds
If
n
a and
n
b are real numbers,
n
a ⋅ n b = n ab
n
a n a
=
if b ≠ 0
b
b
n
Martin-Gay, Developmental Mathematics
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Multiplying and Dividing Surds
Example 6: Simplify the following.
Martin-Gay, Developmental Mathematics
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Rationalizing the Denominator
Many times it is helpful to rewrite a quotient with the
surd confined to ONLY the numerator.
If we rewrite the expression so that there is no surd in
the denominator, it is called rationalizing the
denominator.
This process involves multiplying the quotient by a
form of 1 that will eliminate the surd in the
denominator.
Martin-Gay, Developmental Mathematics
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Rationalizing the Denominator
Example 7: Rationalize the following.
Martin-Gay, Developmental Mathematics
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Conjugates
Many rational quotients have a sum or
difference of terms in a denominator, rather
than a single surd.
In that case, we need to multiply by the
conjugate of the numerator or denominator
(which ever one we are rationalizing).
The conjugate uses the same terms, but the
opposite operation (+ or −).
Martin-Gay, Developmental Mathematics
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Rationalizing the Denominator
Example 8: Rationalize the following.
Martin-Gay, Developmental Mathematics
!26
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