Given that 7 × 49 = 343, find
70 × 49 =
7 × 4.9 =
343 ÷ 7\
70 × 49 =
7 × 4.9 =
343 ÷ 7\
Worked Solution & Answer
343 × 10 = 3430, 343 ÷ 10 = 34.3, 343 ÷ 7 = 49 B1
Answer: 3430, 34.3, 49
Simplifying algebra — Notes & Worked Examples Show / Hide
Explanation:
To simplify by collecting like terms, group terms that have the exact same letter and power (note that x and x2 are not like terms). Always keep the + or − sign immediately in front of each term attached to that term when grouping.
When multiplying algebraic terms, multiply the numbers in front (coefficients) first, then write the letters together in alphabetical order (remembering m × m = m2). When dividing, divide the coefficients and cancel common letters.
Example 1: Simplify:
(a) 5a + 3b − 2a + 4b (b) 4x2 − 3x + 2x2 − 5x
Solution:
(a) Group the a terms and b terms with their signs:
• (5a − 2a) + (3b + 4b) = 3a + 7b
(b) Group x2 terms and x terms separately:
• (4x2 + 2x2) + (−3x − 5x) = 6x2 − 8x
Example 2: Simplify:
(a) 4p × 3q (b) 5m × 3m (c) 18y2 ÷ 6y
Solution:
(a) 4 × 3 = 12, and p × q = pq → 12pq
(b) 5 × 3 = 15, and m × m = m2 → 15m2
(c) 18y26y: 18 ÷ 6 = 3 and y2 ÷ y = y → 3y
Work out
16 × 25 × 4\
16 × 25 × 4\
Worked Solution & Answer
16 × (25 × 4) = 16 × 100 = 1600 B1
Answer: 1600
Simplify fully 8a + 5b + 6a - 2b
Answer
Worked Solution & Answer
Collect like terms: M1
8a + 6a = 14a
5b - 2b = 3b
14a + 3b A1
8a + 6a = 14a
5b - 2b = 3b
14a + 3b A1
Answer: Answer 14a + 3b
(a) Simplify 8h³ + 14h³ – 2h³ (1) (b) Simplify (9y + 12y) ÷ 3 (1)
(a) Simplify 8h3 + 14h3 - 2h3 [1 mark]
(b) Simplify (9y + 12y) ÷ 3 [1 mark]
Worked Solution & Answer
(a) Simplify 3 × 4t (1) (b) Simplify 8a – 3a + 2a (1)
(a) Simplify 3 × 4t [1 mark]
(b) Simplify 8a - 3a + 2a [1 mark]
Worked Solution & Answer
(a)
3 × 4t = 12t B1
Answer (a): 12t
(b)
(8 – 3 + 2)a = 7a B1
Answer (b): 7a
In the diagram above, the expression in each block in the top two rows is found by using the following rule:
*The expression in any block is the sum of the expressions in the two blocks below it.*
Some expressions are already shown.
Use the rule to fill in the empty blocks in the diagram.
You must simplify your answers.
*The expression in any block is the sum of the expressions in the two blocks below it.*
Some expressions are already shown.
Use the rule to fill in the empty blocks in the diagram.
You must simplify your answers.
Fill in the three empty blocks with simplified expressions.
Worked Solution & Answer
1. Middle left block = (2x + 5) + (3x − 2) = 5x + 3. M1
2. Bottom right block: (3x − 2) + Block = 7x + 4 ⇒ Block = (7x + 4) − (3x − 2) = 4x + 6. M1
3. Top block = (5x + 3) + (7x + 4) = 12x + 7. A1
2. Bottom right block: (3x − 2) + Block = 7x + 4 ⇒ Block = (7x + 4) − (3x − 2) = 4x + 6. M1
3. Top block = (5x + 3) + (7x + 4) = 12x + 7. A1
Answer: Middle left block: 5x + 3; Bottom right block: 4x + 6; Top block: 12x + 7
Work out
(a) Simplify 3f × 5g (1) (b) Simplify t × t (1) (c) Simplify (2n + 6n) / 2 (1)
(a) Simplify 3f × 5g (1) (b) Simplify t × t (1) (c) Simplify (2n + 6n) / 2 (1)
(a) Simplify 3f × 5g [1 mark]
(b) Simplify t × t [1 mark]
(c) Simplify 2n + 6n2 [1 mark]
n
Worked Solution & Answer
(a)
3 × 5 × f × g = 15fg B1
Answer (a): 15fg
(b)
t × t = t² B1
Answer (b): t2
(c)
8n ÷ 2 = 4n B1
Answer (c): 4n n
Expand and simplify (x - 4)(2x + 3)2
Answer
Worked Solution & Answer
First expand (2x + 3)2: M1
(2x + 3)2 = 4x2 + 12x + 9 M1
Now multiply by (x - 4):
(x - 4)(4x2 + 12x + 9) = x(4x2 + 12x + 9) - 4(4x2 + 12x + 9) M1
= 4x3 + 12x2 + 9x - 16x2 - 48x - 36
= 4x3 - 4x2 - 39x - 36 A1
(2x + 3)2 = 4x2 + 12x + 9 M1
Now multiply by (x - 4):
(x - 4)(4x2 + 12x + 9) = x(4x2 + 12x + 9) - 4(4x2 + 12x + 9) M1
= 4x3 + 12x2 + 9x - 16x2 - 48x - 36
= 4x3 - 4x2 - 39x - 36 A1
Answer: Answer 4x3 - 4x2 - 39x - 36
Work out
(a) (i) 3a × 4
(ii) b × b × b × b
(iii) c2 × c4
(b) Factorise.
9 - 6y
(a) (i) 3a × 4
(ii) b × b × b × b
(iii) c2 × c4
(b) Factorise.
9 - 6y
(a) 3a × 4 [1 mark]
(b) b × b × b × b [1 mark]
(c) c2 × c4 [1 mark]
(b) Factorise.
9 - 6y [1 mark]
9 - 6y [1 mark]
Worked Solution & Answer
(a)
3 × 4 × a = 12a. B1
Answer (a): 12a
(b)
b multiplied by itself 4 times is b⁴. B1
Answer (b): b4
(c)
c²⁺⁴ = c⁶. B1
Answer (c): c6
(b)
HCF of 9 and 6y is 3 ⇒ 3(3 − 2y). B1
Answer (b): 3(3 - 2y)
Work out
(a) (i) 5r - 7t - 3r + 2t
(ii) a × a × a × a × a
(iii) 7b5 ÷ b
(b) Factorise.
4a - 12b
(a) (i) 5r - 7t - 3r + 2t
(ii) a × a × a × a × a
(iii) 7b5 ÷ b
(b) Factorise.
4a - 12b
(a) 5r - 7t - 3r + 2t [2 marks]
(b) a × a × a × a × a [1 mark]
(c) 7b5 ÷ b [1 mark]
(b) Factorise.
4a - 12b [1 mark]
4a - 12b [1 mark]
Worked Solution & Answer
(a)
5r − 3r = 2r and −7t + 2t = −5t M1
⇒ 2r − 5t. A1
⇒ 2r − 5t. A1
Answer (a): 2r - 5t
(b)
a multiplied by itself 5 times is a⁵. B1
Answer (b): a5
(c)
7b⁵ ÷ b¹ = 7b⁵⁻¹ = 7b⁴. B1
Answer (c): 7b4
(b)
Take out the highest common factor 4: 4(a − 3b). B1
Answer (b): 4(a - 3b)