# Exercise 12.1

Question 1

Evaluate

(i) 3^{−2} (ii) (−4)^{−2} (iii)

Sol :

(i)

(ii)

(iii)

Question 2

Simplify and express the result in power notation with positive exponent.

(i) (ii)

(iii) (iv)

(v)

Sol :

(i) (−4)^{5} ÷ (−4)^{8} = (−4)^{5 − 8} (*a*^{m} ÷ *a*^{n} = *a*^{m}^{ − }^{n})

= (− 4)^{−3}

(ii)

(iii)

(iv) (3^{− 7} ÷ 3^{−10}) × 3^{−5} = (3^{−7 − (−10)}) × 3^{−5} (*a*^{m} ÷ *a*^{n} = *a*^{m }^{− }^{n})

= 3^{3} × 3^{−5}

= 3^{3 + (− 5)} (*a*^{m} × *a*^{n} = *a*^{m}^{ + }^{n})

= 3^{−2}

(v) 2^{−3} × (−7)^{−3} =

Question 3

Find the value of.

(i) (3^{0} + 4^{−1}) × 2^{2} (ii) (2^{−1} × 4^{−1}) ÷2^{−2}

(iii) (iv) (3^{−1} + 4^{−1} + 5^{−1})^{0}

(v)

Sol :

(i)

(ii) (2^{−1} × 4^{−1}) ÷ 2^{− 2 }= [2^{−1} × {(2)^{2}}^{− 1}] ÷ 2^{− 2}

= (2^{− 1} × 2^{− 2}) ÷ 2^{− 2}

= 2^{−1+ (−2)} ÷ 2^{−2} (*a*^{m} × *a*^{n} = *a*^{m}^{ + }^{n})

= 2^{−3} ÷ 2^{−2}

= 2^{−3} ^{− (−2)} (*a*^{m} ÷ *a*^{n} = *a*^{m}^{ − }^{n})

= 2^{−3 + 2} = 2^{ −1}

(iii)

(iv) (3^{−1} + 4^{−1} + 5^{−1})^{0}

= 1 (*a*^{0} = 1)

(v)

Question 4

Evaluate (i) (ii)

Sol :

(i)

(ii)

Question 5

Find the value of *m* for which 5^{m} ÷5^{−3} = 5^{5}.

Sol :

5^{m} ÷ 5^{−3} = 5^{5}

5^{m} ^{− (− 3)} = 5^{5} (*a*^{m} ÷ *a*^{n} = *a*^{m}^{ − }^{n})

5^{m}^{ + 3} = 5^{5}

Since the powers have same bases on both sides, their respective exponents must be equal.

*m* + 3 = 5

*m* = 5 − 3

*m* = 2

Question 6

Evaluate (i) (ii)

Sol :

(i)

(ii)

Question 7

Simplify. (i) (ii)

Sol :

(i)

(ii)