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UNIT 1A LESSON 3
OPERATIONS WITH
RATIONALS & RADICALS
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Addition & Subtraction of rational expressions requires a common denominator
*Rational Expressions
(π )3π₯8 (π)
+(π )5 π₯6(π)
β (π )π₯3(π)
3π₯8 +
5 π₯6 β
π₯3
9π₯24 +
20π₯24 β
8 π₯24
ΒΏ21π₯24
ΒΏ7 π₯8
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Addition & Subtraction of rational expressions requires a common denominator
*Rational Expressions34 +
π₯6
3 (3)4 (3)
+π₯ (2)6(2)
912+
2π₯12
2π₯+912
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Addition & Subtraction of rational expressions requires a common denominator
*Rational Expressions2π₯ +
4π₯β3
2(πβπ)π₯ (π βπ)
+ 4 ππ (π₯β3)
2 π₯β6π₯ (π₯β3)+
4 π₯π₯ (π₯β3)
6 π₯β6π₯ (π₯β3)6(π₯β1)π₯ (π₯β3)
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( π₯β54 )Γ·( π₯+28 )
To divide rational expressions we multiply by the reciprocal.
π₯β54 Γ π
π+π
8(π₯β5)4 (π₯+2)
2(π₯β5)(π₯+2)
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21
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To divide rational expressions we multiply by the reciprocal.π₯ββ23π
π+βπ
π₯ββ23
Γ π+βππ
(π₯ββ2)(π₯+β2)15
π₯2β215
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Consider this division of a compound fraction(π+π )(πβπ)
Γππ
ππ(π)(π+π )
(πβπ)Γ·π
[ π+ππβπ ]π
[ π+ππβπ ]π
(π+π )π(πβπ)
ππ
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π ππ +
ππ
π
(π)2π₯5 (4 )
+5(1)4(5)
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8 π₯20 +
520
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8 π₯+5203
8 π₯+520 Γ 13=
8 π₯+560
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ππβπ+
ππ
ππ
2 ππ (π₯β5)
+3(π βπ)π₯ (π βπ)
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5π₯β15 π₯ (π₯β5) Γ
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5ΒΏΒΏ
ΒΏΒΏ2π₯+3 π₯β15π₯ (π₯β5) Γ·25
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πβπππ βπ
π₯2π₯ β
1π₯
π₯β1
π₯2β1π₯
π₯β1
π₯2β1π₯ (π₯β1)
(π₯β1)(π₯+1)π₯(π₯β1)
π₯+1π₯
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β4h hπ₯ (π₯+h)
β4 π₯ (π₯+h)
ππ+π β
ππ
π
4 π₯π₯ (π₯+h)
β 4 (π₯+h)π₯ (π₯+h)
h
4 π₯β4 π₯β4h hπ₯(π₯+h)
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* RATIONALIZING NUMERATORS AND DENOMINATORS
*The ability to simplify an expression by rationalizing it is important in problems involving limits. *Recall that the conjugate of any expression of
the form is , and vice versa.
*Rationalizing often involves multiplying the numerator and denominator by the conjugate of the expression that is being rationalized.
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Rationalize the denominator Rationalize the numerator
5β2
3β34
5β2Γ βπ
βπ3β34Γ βπ
βπ
5β22
94 β3
2 ππ π πππ‘πππππππ’ππππ 9 ππ ππππ‘πππππππ’ππππ
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Rationalize the denominator Rationalize the numerator
51+β2
β5β32
51+β2
Γπββππββπ
5β5β21β2
β5β32
Γ βπ+πβπ+π
5β5β2β1
β5+5β2
5β92 (β5+3 )
β42 (β5+3 )
β2β5+3
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Rationalize the denominator of the expression
2βπ₯βπ₯β2
Γ βπ+πβπ+π
2βπ₯ (βπ₯+2)π₯β4
2π₯+4βπ₯π₯β4
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Rationalize the numerator of the expression
2βπ₯βπ₯β2
Γ βπβπ
2π₯π₯β2βπ₯
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Rationalize the numerator of the expression
βπ₯β2π₯β4 Γ
βπ+πβπ+π
π₯β4(π₯β4)(βπ₯+2)
1βπ₯+2
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Rationalize the numerator of the expression
π₯β4(π₯β4)(π₯+4)(βπ₯+5+3)
1(π₯+4 )(βπ₯+5+3)
βπ₯+5β3π₯2β16
Γ βπ+π+πβπ+π+π
π₯+5β9(π₯β4)(π₯+4)(βπ₯+5+3)1
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