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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

(𝟑 )3𝑥8 (𝟑)

+(𝟒 )5 𝑥6(𝟒)

− (𝟖 )𝑥3(𝟖)

3𝑥8 +

5 𝑥6 −

𝑥3

9𝑥24 +

20𝑥24 −

8 𝑥24

¿21𝑥24

¿7 𝑥8

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Addition & Subtraction of rational expressions requires a common denominator

*Rational Expressions34 +

𝑥6

3 (3)4 (3)

+𝑥 (2)6(2)

912+

2𝑥12

2𝑥+912

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Addition & Subtraction of rational expressions requires a common denominator

*Rational Expressions2𝑥 +

4𝑥−3

2(𝒙−𝟑)𝑥 (𝒙 −𝟑)

+ 4 𝒙𝒙 (𝑥−3)

2 𝑥−6𝑥 (𝑥−3)+

4 𝑥𝑥 (𝑥−3)

6 𝑥−6𝑥 (𝑥−3)6(𝑥−1)𝑥 (𝑥−3)

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( 𝑥−54 )÷( 𝑥+28 )

To divide rational expressions we multiply by the reciprocal.

𝑥−54 × 𝟖

𝒙+𝟐

8(𝑥−5)4 (𝑥+2)

2(𝑥−5)(𝑥+2)

2

21

1

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To divide rational expressions we multiply by the reciprocal.𝑥−√23𝟓

𝒙+√𝟐

𝑥−√23

× 𝒙+√𝟐𝟓

(𝑥−√2)(𝑥+√2)15

𝑥2−215

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Consider this division of a compound fraction(𝟕+𝟐 )(𝟓−𝟏)

×𝟏𝟑

𝟗𝟑(𝟒)(𝟕+𝟐 )

(𝟓−𝟏)÷𝟑

[ 𝟕+𝟐𝟓−𝟏 ]𝟑

[ 𝟕+𝟐𝟓−𝟏 ]𝟑

(𝟕+𝟐 )𝟑(𝟓−𝟏)

𝟑𝟒

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𝟐 𝒙𝟓 +

𝟏𝟒

𝟑

(𝟒)2𝑥5 (4 )

+5(1)4(5)

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8 𝑥20 +

520

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8 𝑥+5203

8 𝑥+520 × 13=

8 𝑥+560

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𝟐𝒙−𝟓+

𝟑𝒙

𝟐𝟓

2 𝒙𝒙 (𝑥−5)

+3(𝒙 −𝟓)𝑥 (𝒙 −𝟓)

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5𝑥−15  𝑥 (𝑥−5) ×

125

5¿¿

¿¿2𝑥+3 𝑥−15𝑥 (𝑥−5) ÷25

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𝒙−𝟏𝒙𝒙 −𝟏

𝑥2𝑥 −

1𝑥

𝑥−1

𝑥2−1𝑥

𝑥−1

𝑥2−1𝑥 (𝑥−1)

(𝑥−1)(𝑥+1)𝑥(𝑥−1)

𝑥+1𝑥

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−4h  h𝑥 (𝑥+h)

−4  𝑥 (𝑥+h)

𝟒𝒙+𝒉 −

𝟒𝒙

𝒉

4 𝑥𝑥 (𝑥+h)

− 4 (𝑥+h)𝑥 (𝑥+h)

h

4 𝑥−4 𝑥−4h  h𝑥(𝑥+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

5√2

3√34

5√2× √𝟐

√𝟐3√34× √𝟑

√𝟑

5√22

94 √3

2 𝑖𝑠𝑎 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟 9 𝑖𝑠 𝑎𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟

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Rationalize the denominator Rationalize the numerator

51+√2

√5−32

51+√2

×𝟏−√𝟐𝟏−√𝟐

5−5√21−2

√5−32

× √𝟓+𝟑√𝟓+𝟑

5−5√2−1

−5+5√2

5−92 (√5+3 )

−42 (√5+3 )

−2√5+3

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Rationalize the denominator of the expression

2√𝑥√𝑥−2

× √𝒙+𝟐√𝒙+𝟐

2√𝑥 (√𝑥+2)𝑥−4

2𝑥+4√𝑥𝑥−4

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Rationalize the numerator of the expression

2√𝑥√𝑥−2

× √𝒙√𝒙

2𝑥𝑥−2√𝑥

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Rationalize the numerator of the expression

√𝑥−2𝑥−4 ×

√𝒙+𝟐√𝒙+𝟐

𝑥−4(𝑥−4)(√𝑥+2)

1√𝑥+2

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Rationalize the numerator of the expression

𝑥−4(𝑥−4)(𝑥+4)(√𝑥+5+3)

1(𝑥+4 )(√𝑥+5+3)

√𝑥+5−3𝑥2−16

× √𝒙+𝟓+𝟑√𝒙+𝟓+𝟑

𝑥+5−9(𝑥−4)(𝑥+4)(√𝑥+5+3)1

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