8 Math Exponents And Power

Finish Line & Beyond

Exponents and Power 1. Numbers with negative exponents obey the following laws of exponents. (a) am × an = amn (b) am ÷ an = am–n (c) (am)n = amn (d) am × bm = (ab)m (e) a0 = 1 (f) am/ bm =(a/b)m 2. Very small numbers can be expressed in standard form using negative exponents.

Exercise 1 1. Evaluate. (i) 3–2 Answer: 3–2 =

1 1 = 3² 9

(ii) (– 4)– 2 Answer: (– 4)– 2 =

(iii)(

1 1 = − 4² 16

1 -2 1 -2 1 -2 ) +( ) +( ) 2 3 4

Answer: (

1 -2 1 -2 1 -2 ) +( ) +( ) 2 3 4

= 2²+3²+4² = 4+9+16 = 29 2. Simplify and express the result in power notation with positive exponent. (i) (– 4)5 ÷ (– 4)8 Answer: am ÷ an = am-n So, (– 4)5 ÷ (– 4)8 =(-4)5-8 = (-4)-3

1 1 )³ = − 4 64 1 (ii) ( )² 2³ 1 Answer: ( )² 2³ =(

(am)n = amn

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Finish Line & Beyond 1 )² 2³ 1 = 1/26 = 64 Hence, (

(iii) (-3) × (

5 ) 3

=3 × 5 ÷ 3 (the power on 3 is even so -3 = 3) =5=625 Another way of solving is using am ÷ an = am-n So, -3 ÷ 3=-3°=1 (iv) (3–7 ÷ 3–10) × 3–5 Answer: Using am ÷ an = am-n (3–7 ÷ 3–10) = 3-7+10 = 3 Hence, (3–7 ÷ 3–10) × 3–5 =3 × 3–5 Now, am × an=am+n Hence, =3 × 3–5 =33-5=3-2 =

1 1 = 3² 9

(v) 2– 3 × (–7)–3 Answer: 2– 3 × (–7)–3

1 1 × 2³ − 7³ 1 1 1 = × = 8 243 1944

=

3. Find the value of. (i) (3° + 4–1) × 22 Answer: (3° + 4–1) × 22 = (1+ =

1 )× 4 4

5 × 4= 5 4

(ii) (2–1 × 4–1) ÷ 2–2

1 1 1 × ÷ 2 4 2× 2 1 1 1 = × × 4= 2 4 2

Answer:

(iii) (3–1 + 4–1 + 5–1)0

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Finish Line & Beyond

Answer: aº=1 Hence, (3–1 + 4–1 + 5–1)0 = 1 (iv)((

− 2 -2 ) )² 3

Answer: (am)n =amn Hence, ((

− 2 -4 ) 3 3 =( ) − 2 81 = 16

− 2 -2 ) )² 3

=(

4. Evaluate (i) 8-1 × 4³ ÷ 2-4 Answer:

=

1 1 × 64 ÷ 8 16

1 × 64 × 16 = 128 8

(ii) (5–1 × 2–1) × 6–1 Answer:

=

1 1 1 1 × × = 5 2 6 60

5. Find the value of m for which 5m ÷ 5–3 = 55 Answer: am = am-n Here, m-n = 5 and n = -3 So, m = 5+(-3)=2 6. Evaluate (i) ((

1 -1 1 -1 -1 ) -( ) ) 3 4

Answer: (3-4)-1 = (-1)-1 =-1 (ii) (

5 -7 8 -4 ) × ( ) 8 5

Answer: =(

8 7 5 4 ) × ( ) 5 8

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Finish Line & Beyond =87/84 × 54/57 = 8³ ×

1 512 = 5³ 125

7. Simplify. (i) (25 × t-4) ÷ (5-3 × 10 × t-8) ( t

≠ 0)

Answer: (25 × t-4) ÷ (5-3 × 10 × t-8) =(5² × t-4) ÷ (5-3 × 5 × 2 × t-8) = 52+2 × t-4+8 ÷ 2 = 5 × t ÷ 2 (ii) (3-5 × 10-5 × 125) ÷ (5-7 × 6-5) Answer: (3-5 × 10-5 × 125) ÷ (5-7 × 6-5) = (3-5 × 5-5 × 2-5 × 5) ÷ (5-7 × 6-5) = (3-5 × 5-5+3 × 2-5) ÷ (5-7 × 6-5) = (3-5 × 5-2 × 2-5) ÷ (5-7 × 3-5 × 2-5) = (3-5+5 × 5-2+7 × 2-5+5) = (30 × 55 × 20) =1 × 3125=3125

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