Simplify √48, giving your answer in the form a √b with b simplified.
(2)
8
Simplify √75 in exact form.
(2)
9
Simplify √32 in exact form.
(2)
10
Write √98 in the form a √b with b simplified.
(2)
11
Find each cube root exactly.
(a)cube root of 343(1)
(b)cube root of 512(1)
12
Evaluate the powers exactly.
(a)25(1)
(b)53(1)
13
Multiply the surds and simplify: √5 x √20.
(2)
14
Simplify and combine: √28 + √63. Give the exact form.
(2)
15
A square rug has an area of 45 m2. Work out the exact length of one side of the rug, giving your answer in the form a √b m.
(3)
16
Express √80 + √45 in the form k √5. Find the value of k, showing your working.
(3)
17
A cube-shaped storage box has a volume of 729 cm3. Work out the exact length of one edge of the box.
(3)
18
Mia says that √9 + √16 is equal to √25. Use exact values to show whether Mia is correct, and state the correct value of √9 + √16.
(3)
19
A square tile has a side length equal to the cube root of 64 cm.
(a)Find the side length of the tile.(1)
(b)Find the area of the tile.(2)
20
A cuboid-shaped tank has a square base with area 20 cm2. The height of the tank is √5 cm. Find the exact volume of the tank in simplest surd form, stating the units.
(3)
Mark scheme · KS3.M-N15D Roots, Powers and Exact Values: Fluency and Exam Drill
Question 1
B1 11 cao
Answer: 11
Question 2
B1 6 cao
Answer: 6
Question 3
B1 81 cao
Answer: 81
Question 4
B1 64 cao
Answer: 64
Question 5
B1 13 cao
Answer: 13
Question 6
B1 8 cao
Answer: 8
Question 7
M1 factor 48 as 16 x 3 or identify 16 as largest square factor
A1 4 √3 cao
Answer: 4 √3
Question 8
M1 factor 75 as 25 x 3 or identify 25 as largest square factor
A1 5 √3 cao
Answer: 5 √3
Question 9
M1 factor 32 as 16 x 2 or identify 16 as largest square factor
A1 4 √2 cao
Answer: 4 √2
Question 10
M1 factor 98 as 49 x 2 or identify 49 as largest square factor
A1 7 √2 cao
Answer: 7 √2
Question 11
(a) B1 7 cao
(a) Answer: 7
(b) B1 8 cao
(b) Answer: 8
Question 12
(a) B1 32 cao
(a) Answer: 32
(b) B1 125 cao
(b) Answer: 125
Question 13
M1 use √5 x √20 = √100 or simplify √20 to 2 √5 first
A1 10 cao
Answer: 10
Question 14
M1 simplify √28 to 2 √7 or √63 to 3 √7, or show common surd factor
A1 5 √7 cao
Answer: 5 √7
Question 15
M1 use side = √area, i.e. side = √45
M1 factor 45 as 9 x 5 or identify 9 as largest square factor
A1 3 √5 m cao
Answer: 3 √5 m
Question 16
M1 simplify √80 to 4 √5
M1 simplify √45 to 3 √5
A1 7 √5 cao, so k = 7
Answer: 7 √5, so k = 7
Question 17
M1 use edge = cube root of volume
M1 recognise 729 = 9 x 9 x 9 (trial or systematic check)
A1 9 cm cao
Answer: 9 cm
Question 18
M1 evaluate √9 = 3 and √16 = 4, giving sum 7
M1 evaluate √25 = 5 and compare with 7
A1 conclusion that Mia is incorrect since 7 ≠ 5; correct value is 7
Answer: Mia is incorrect. √9 + √16 = 3 + 4 = 7, but √25 = 5.