Exercise 2.3
Surds
Chapter Sections
Introduction to Surds
Simplifying Surds
Adding and Subtracting Surds
Multiplying and Dividing Surds
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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.
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Square Roots
The principal (positive) square root is noted
as
a
The negative square root is noted as
− a
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Square Roots
Example 1: Simplify the following.
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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.
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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.
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Cube Roots
Example 2: Simplify the following.
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Simplifying Surds
Product Rule for Surds
If
a and
b are real numbers,
ab = a ⋅ b
a
a
=
if
b
b
b ≠0
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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
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Simplifying Surds
Example 3: Simplify the following.
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Simplifying Surds
Example 4: Simplify the following.
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Adding and Subtracting Surds
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
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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.
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Adding and Subtracting Surds
Example
3+7 3 = 8 3
10 2 − 4 2 = 6 2
5+ 3
Can not simplify
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Adding and Subtracting Surds
Example 5: Simplify the following.
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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
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Multiplying and Dividing Surds
Example 6: Simplify the following.
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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.
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Rationalizing the Denominator
Example 7: Rationalize the following.
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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 −).
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Rationalizing the Denominator
Example 8: Rationalize the following.
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