Year 8: Algebraic Fractions

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 30th August 2015

Are these algebraic steps correct?

40 - x3

Fail Win!

(Click your answer)

= x + 4 40 3

= 2x + 4

2(4 – 2x) = 3x - 2 2(4) = 5x - 2

Fail Win!

√2−𝑥=2 𝑥+3 √2=3 𝑥+2Fail Win!

Starter

Are these algebraic steps correct?

Fail Win!

(Click your answer)

𝑎2𝑏𝑎+𝑏

𝑎𝑏𝑏

Starter

Are these algebraic steps correct?

(x+3)2

Fail Win!

(Click your answer)

x2 + 32

(3x)2

Fail Win!

32x2 9x2

Starter

To cancel or not to cancel, that is the question?

y2 + x2 + x

s(4 + z)s √𝑥2+2=𝑦+2

(2x+1)(x – 2)x – 2

pq(r+2) + 1pq

Fail Win! Fail Win!

Fail Win!

Fail Win!

Fail Win!

1 + r2

Fail Win!

- 1

(Click your answer)

Starter

What did we learn?

Bro Tip #1: You can’t add or subtract a term which is ‘trapped’ inside a bracket, fraction or root.

Bro Tip #2: In a fraction, we can only divide top and bottom by something, not add/subtract. (e.g. is not the same as !)

Adding/Subtracting Fractions

What’s our usual approach for adding fractions?

Sometimes we don’t need to multiply the denominators. We can find the Lowest Common Multiple of the denominators.

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Adding/Subtracting Algebraic Fractions

The same principle can be applied to algebraic fractions.

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Bro Tip: Notice that with this one, we didn’t need to times x and x2 together: x2 is a multiple of both denominators.

Further Examples

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Bro Tip: Be careful with your negatives!

“To learn the secret ways of the ninja, add fractions you must.”

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Test Your Understanding

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Exercise 1

Harder Questions

3𝑥

+ 2𝑥−1

=3 (𝑥−1 )+2 𝑥𝑥 (𝑥−1 )

= 5𝑥−3𝑥 (𝑥−1 )?

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If were to add say, then we could use 6 as the denominator because is divisible by both 2 and 3.

This gives us a clue what we could use as a denominator .

We can do a cross-multiplication type thing just as before.

2𝑥

+ 3𝑥+1

=2 (𝑥+1 )+3 𝑥  𝑥(𝑥+1)

= 5 𝑥+2𝑥(𝑥+1)

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𝑥+1𝑥−

𝑥𝑥+1

=(𝑥+1 )2−𝑥2

𝑥 (𝑥+1 )= 2 𝑥+1𝑥 (𝑥+1 )

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Test Your Understanding

1𝑥−1

−3

𝑥+3=1 (𝑥+3 )−3 ( 𝑥−1 )  

(𝑥−1)(𝑥+3)= −2 𝑥+6

(𝑥−1) (𝑥+3)? ?

1+1

𝑥−1=11+1

𝑥−1=

𝑥𝑥−1? ?

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Exercise 2

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Extra Practice

y2

2x3

× = xy2

6z2

4x3

= 3z2

4x

x+13

x+24

= 4(x+1)3(x+2)

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Multiplying and DividingThe same rules apply as with normal fractions.

( 𝑥3

2 )2

=𝑥6

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Test Your Understanding

x2

243x

× = 2x3?

( 𝑥2 𝑦 3𝑧5 )3

=𝒙𝟔𝒚𝟗

𝒛𝟏𝟓?

2𝑥+13

÷𝑦+45

=5 (2 𝑥+1 )3 (𝑦+4 )

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y3

2xy× = xy2

2

x2y

xy× = x2

2y2

x+1x2

xy× = x+1

xy

2xy

zq

= 2qxyz

x+1y

z+1q

= q(x+1)y(z+1)

q2

y+1xq

= q3

x(y+1)

( )xy2

= x2

y4

2

( )2q5

z3= 4q10

z6

2

( )3xy

= 9x2

y2

2

( )3x2y3

2z4= 27x6y9

8z12

3

( )x+13y

= (x+1)2

9y2

2

12

( )x+13y

= (x+1)2

9y2

2

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Exercise 3

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Head Table

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Rear Table

Head To Head

1𝑥

+3𝑥

Answer:

Question 1

𝑥2

+𝑥4

Answer:

Question 2

23+𝑥+19

Answer:

Question 3

𝑥𝑦

+𝑥+1𝑦 2

Answer:

Question 4

1𝑥

+𝑥𝑦

Answer:

Question 5

1𝑧

+1

𝑧+2

Answer:

Question 6

1𝑥

+1𝑦

+1𝑧

Answer:

Question 7

1𝑥÷3

Answer:

Question 8

2÷1

𝑥2

Answer:

Question 9

1𝑥÷1

𝑥2 𝑦

Answer:

Question 10

( 𝑥𝑦 3 )2

Answer:

Question 11