Indices

1. Solve each of the following equations.

2 2
(a) 52𝑥 = 57𝑥−1 (b) 42𝑥+1 = 43𝑥−2 (c) 7𝑥 = 76−𝑥 (d) 32𝑥 = 39𝑥+5

2. Solve each of the following equations.

(a) 2𝑛+1 = 32 (b) 42𝑛 = 256 (c) 2𝑛+2 = 128 (d) 32𝑛+1 = 27
1 1 1 2 −16
(e) 2𝑛−1 = 4 (f) 23𝑛+2 = 128 (g) 5𝑛+1 = 125 (h) 5𝑥 =1

3. Solve each of the following equations.

(a) 2𝑥 = 43 (b) 32𝑥−1 = 27𝑥 (c) 53𝑥−7 = 252𝑥 (d) 3𝑥 = 9𝑥+5

1 𝑥 1
(e) 43𝑥+4 = 84𝑥+12 (f) 252𝑥+1 = 1253𝑥+2 (g) (4) = 64 (h) 45−3𝑥 = 8𝑥+1

1 𝑥+1 2 +3 2 −4 2 −2
(i) 84𝑥−3 = (16) (j) 5𝑥 = 252𝑥 (k) 3𝑥 = 27𝑥 (l) 22𝑥 − 8𝑥 = 0

4. Solve each of the following equations.

(a) 23𝑥 × 4𝑥+1 = 64 (b) 23𝑥+1 × 8𝑥−1 = 128

1
(c) (22−𝑥 )(42𝑥+3 ) = 8 (d) 3𝑥+1 × 92−𝑥 =
27

5. Solve each of the following equations:

272𝑥 32𝑥+1 4𝑥 23𝑥 2𝑥+4 64 272𝑥 32𝑥+1
(a) = (b) = (c) = 1 (d) =
35−𝑥 9𝑥+3 23−𝑥 8𝑥−2 8−𝑥 𝑥 36−𝑥 9𝑥+3
42

6. Solve each of the following pairs of simultaneous equations:

(a) 4𝑥 ÷ 2𝑦 = 16 and 32𝑥 × 9𝑦 = 27 (b) 27𝑥 = 9(3𝑦 ) and 2𝑥 ÷ 8𝑦 = 1

1 1−𝑦
(c) 125𝑥 ÷ 5𝑦 = 25 and 23𝑥 × (8) = 32.

Answers:
1 1
[1] (a) , (b) 3, (c) −3 or 2, (d) − or 5, [2] (a) 4, (b) 2, (c) 5, (d) 1, (e) −1, (f) −3, (g) −4, (h) ±4, [3] (a) 6,
5 2

2 4 1 5
(b) −1, (c) −7, (d) −10, (e) −4 , (f) − , (g) −3, (h) 4 , (i) , (j) 1 or 3, (k) −1 or 4, (l) −0.5 or 2, [4] (a) 0.8,
3 5 3 16

5 1 11 1 3 1 7 3
(b) 1.5, (c) − 3, (d) 8, [5] (a) 0, (b) 3, (c) 0.4, (d) 7, [6] (a) 𝑥 = 6
, 𝑦 = − 3, (b) 𝑥 = 4, 𝑦 = 4, (c) 𝑥 = 6, 𝑦 = 2.

FAISAL MIZAN 1
 Surds

1. Simplify the following surds:

(a) √8 + √18 − 2√2 (b) √75 + 2√12 − √27 (c) √28 + √175 − √63

(d) √1000 − √40 − √90 (e) √512 + √128 + √32 (f) √27 + √243
5
(g) √24 − 3√6 − √216 + √294 (h) √175 + √112 − √28 (i) (√3) .

2. Express the following in the form 𝐴 + 𝐵√𝐶 and hence find 𝐴, 𝐵 and 𝐶:
2 2
(a) (4√5 − 3) (b) (√5 + 2) (c) (1 + √2)(3 − 2√2)
2
(d) (√3 − 1) (e) (1 − √2)(3 + 2√2)

3. Rationalise the denominators:

1 3 9 √2 4 12
a) b) c) d) e) f)
√5 √2 √3 √6 √5 √3

4 10 3 √2 √3 √12
g) h) i) j) k) l)
√12 √8 √8 √32 √15 √156

5 7 1+√5 (√3−1) (3−√2) (14−√7)

m) 2 n) 2 o) p) q) r)
√2 √3 √5 √3 √2 √7

4. Rationalise the denominators and simplify:

1 1 1 4 5 √7
a) 1+ b) 3+ c) 3+ d) 3− e) 2+ f) 2−
√2 √5 √7 √5 √5 √7

2 5 1 8
g) 2− h) 2 i) 2 j)
√3 √3−3 √ √2
3− √ √5
7−

5. Rationalise the denominators and simplify:

2−√3 1+√2 √2+1 √7−√2

a) 2+ b) 3− c)2 d)
√3 √2 √2−1 √7+√2

√5+1 √17−√11 √3−√7 √23+√37

e) 3− f) g) h)
√5 √17+√11 √3+√7 √37−√23

6. Write as a single fraction:

1 1 2 1 2 1
a) + 3−1 b) + 7− 2 c) 4− + 4+
√3+1 √ √7+√2 √ √ √3 √3

Answers:

[1] (a) 3√2, (b) 6√3, (c) 4√7, (d) 5√10, (e) 28√2, (f) 12√3, (g) 0, (h) 7√7, (i) 9√3, [2] (a) 89 − 24√5,

√5 3√2 √3 4√5
(b) 9 + 4√5, (c) −1 + √2, (d) 4 − 2√3, (e) −1 − √2, [3] (a) 5
, (b) 2
, (c) 3√3, (d) 3
, (e) 5
, (f) 4√3,

FAISAL MIZAN 2
 2√3 5√2 3√2 1 √5 √13 5√2 7√3 5+√5 3−√3 −2+3√2
(g) , (h) , (i) , (j) , (k) , (l) , (m) , (n) , (o) , (p) , (q) , (r) −1 + 2√7,
3 2 4 4 5 13 4 6 5 3 2

3−√5 3−√7 −7−2√7 15+10√3 2√3+√2

[4] (a) −1 + √2, (b) 4
, (c) 2
, (d) 3 + √5, (e) −10 + 5√5, (f) 3
, (g) 4 + 2√3, (h) 3
, (i) 10
,

5+4√2 5+3√2 9−2√14 14−√187 −5+√21

(j) 4√7 + 4√5, [5] (a) 7 − 4√3, (b) 7
, (c) 7
, (d) 5
, (e) 2 + √5, (f) 3
, (g) 2
,

30+√851 3√7−√2 12+√3

(h) 7
, [6] (a) √3, (b) 5
, (c) 13
.

FAISAL MIZAN 3