8th Maths : Chapter 1 : Numbers : Exponents and Powers : Exercise 1.6 : Text Book Back Exercises Questions with Answers, Solution

**Exercise 1.6**

** **

**1. Fill in the blanks:**

(i) (−1)^{even integer}
is _____________.** [Answer:1]**

(ii) For a ≠ 0, a^{0}
is ______________.** [Answer:1]**

(iii) 4^{-3}
× 5^{-3} __________.** [Answer:20 ^{−3}]**

**(**iv) (-2)^{-7}
= ____________.** [Answer:−1/128]**

(v) (-1/3)^{-5
}= ____________.** [Answer: −243]**

2.
**Say True or False:**

(i) If 8* ^{x} *= 1/64, the value of

(ii) The simplified form of
(256)^{−1/4} × 4^{2} is ¼.**[Answer: False]**

(iii) Using
the power rule, (3^{7})^{-2} = 3^{5}.**[Answer: False]**

(iv) The
standard form of 2 × 10^{–4} is 0.0002.

(v) The scientific
form of 123.456 is 1.23456×10^{−}^{2}. **[Answer: False]**

3. Evaluate:

**(i) (1/2) ^{3}**

**Solution:**

(1/2)^{3} = 1^{3}/2^{3} = 1 / [2 × 2 × 2]
= 1 / 8

**(ii) (1/2) ^{−5}. **

**Solution:**

(1/2)^{−5 }= 1^{−5 }/ 2^{−5} = 1 / 2^{−5}
= 2^{5} = 2 × 2 × 2 × 2 × 2 = 32

**(iii) (−5/6) ^{−3}**

**Solution:**

(−5/6)^{−3} = (−5)^{−3 }/ 6^{−3} = 6^{3
}/ (−5)^{−3} = ( (6 × 6 × 6) / (−5 × −5 × −5) ) = − 216/125

**(iv) (2 ^{−5} **

**Solution:**

(2^{−5} × 2^{7}) ÷ 2^{−2}** **= (2^{−5+7})
÷ 2^{−2}

= 2^{2} ÷ 2^{−2}

= 2^{2+2}

= 2^{4} = 16

**(v) (2 ^{−1} **×

**Solution:**

(2^{−1} × 3^{−1})** ÷ **6^{−2} = (2 ×
3)^{−1} **÷ 6 ^{−2}**

= (6)^{−1} **÷** 6^{−2}

= (6)^{−1 – (−2) }= 6^{1 }= 6

4. Evaluate:

**Solution:**

(2/5)^{4}
× (2/5)^{2} = (2/5)^{4+2}
= (2/5)^{6}

**(ii) (4/5) ^{−2} ÷ (4/5)^{−3}**

**(iii) 2 ^{7} **×

**5. Evaluate: (i) (5 ^{0} +6^{-1})
× 3^{2} (ii) (2^{-1} + 3^{-1} ) ÷ 6^{-1 }(iii) (
3^{-1} + 4^{-2} + 5^{-3})^{0}**

**Solution:**

** (i) (5 ^{0} +6^{−1})
× 3^{2}**

5^{0}
× 3^{2} + 6^{−1} × 3^{2} = (1 × 9) + ( {1 / [2×3]} × 3^{2})

= 9 + ( 1/6
× 9 ) = 9 + 3/2

= [18 + 3] /
2 = 21/2

**(ii) (2 ^{−1} + 3^{−1}) ÷ 6^{−1}**

(2^{−1} + 3^{−1}) ÷ 6^{−1}
= (1/2 + 1/3) **÷ **6^{−1}

= ( [3 + 2]
/ 6) **÷ **6^{−1 } = (5/6) **÷
**6^{−1 }= 5/6 × 6 = 5

**(iii) (3 ^{−1} + 4^{−2} + 5^{−3})^{0}**

(3^{−1} + 4^{−2} + 5^{−3})^{0
}= 1

[∵ *a*^{0} = 1
where *a* ≠ 0]

** **

**6. Simplify: (i) (3 ^{2})^{3}
× (2×3^{5})^{–2} × (18)^{2}**

** **

**Solution: **

**(i) (3 ^{2})^{3} **×

(3^{2})^{3}
× (2 × 3^{5})^{−2} × (18)^{2} = 3^{2x3} × [1 /
(2 × 3^{5})^{2} ] × 18^{2}

= 3^{6}
× [1 / 2^{2}×(3^{5})^{2} ] × 18^{2 }= 3^{6}
× [ 1 / 2^{2}×3^{10} ] ×
(2×3^{2})^{2}

= 3^{6}
× [ 1 / 2^{2}×3^{10} ] ×
2^{2} × 3^{2×2} = 3^{6}
× [ 1 / 2^{2}×3^{10 }] ×^{ }2^{2} × 3^{4}

= [3^{6+4}
× 2^{2} ] / [2^{2} × 3^{10}
] = [3^{10} × 2^{2} ] / [ 2^{2} × 3^{10 }] = 1

**(ii) [ 9 ^{2} **×

[ 9^{2}
× 7^{3} × 2^{5} ] / 84^{3} = [ (3^{2})^{2}
× 7^{3} × 2^{5} ] / [ (2^{2} × 3 × 7)^{3} ] = [
3^{2x2} × 7^{3} × 2^{5} ] / [ 2^{2x3}x 3^{3}
× 7^{3} ] = [ 3^{4 × }7^{3} × 2^{5} ] / [ 2^{6
× }3^{3} × 7^{3} ]

= 3^{4−3}
× 7^{3−3} × 2^{5−6} = 3^{1} × 7^{0} × 2^{−1}

= 3 × 1 × 2^{−1}
= 3 × (1/2) = 3/2

**(iii) [ 2 ^{8} **×

**Solution:**

[2^{8}
× 2187] / [3^{5} × 32] = [2^{8} × 3^{7} ] / [ 3^{5}
× 2^{5}]

= 2^{8−5}
× 3^{7−5}

= 2^{3}
× 3^{2} = 8 × 9 = 72

**7. Solve for x: **

**(i) 2 ^{2x−1} / 2^{x}^{+2}
= 4**

**Solution:**

2^{2x−1}
^{–(x+2)} = 2^{2}

2^{2x−1}
^{–x−2)} = 2^{2}

2^{x}^{−3}
= 2^{2}

Equating the
powers of the same base 2.

*x*−3 = 2

*x* – 3 + 3 = 2 + 3

*x* = 5

**(ii) [ 5 ^{5} × 5^{−4} × 5^{x} ] / 5^{12}
= 5^{−5} **

[ 5^{5} × 5^{−4} × 5^{x} ] / 5^{12}
= 5^{−5}

^{}

⇒ [ 5^{5 −4 + x} ]* ^{ }*/ 5

⇒ 5^{1+ x }/ 5^{12} = 5^{−5}

⇒ 5^{1+ x − 12} = 5^{−5}

⇒ 5^{ x −11} = 5^{−5}

Equating the powers of same base 5.

*x* − 11 = −5

*x* —11 + 11 = −5 + 11

*x* = 6

** **

**8. Expand using exponents: (i) 6054.321
(ii) 897.14**

**(i) 6054.321** = (6 × 1000) + (0 × 100) + (5 × 10) + (4 × 10^{0}) +
3/10 + 2/100 + 1/1000

= (6 × 10^{3}) + (5 × l0^{1}) + (4 × 10^{0})
+ ( 3 × 1/10) + ( 2 × 1/100) + (1 × 1/1000)

= (6 × 10^{3}) + (5 × l0^{1}) + (4 × 10^{0})
+ ( 3 × 10^{−1}) + ( 2 × 10^{−2}) + (1 × 10^{−3})

**(ii) 897.14** = (8 × 100) + (9 × 10) + (7 × 10^{0}) + l/10 + 4/100

= (8 × 10^{2}) +
(9 × 10^{1}) + (7 × 10^{0}) + (l × 1/10) + (4 × 1/100)

= (8 × 10^{2}) +
(9 × 10^{1}) + (7 × 10^{0}) + (l × 10^{−1}) + (4 × 10^{−2})

** **

**9. Find the number in standard form for
the following expansions:**

**(i) 8****×****10 ^{4} **

**(ii) 5****×****10 ^{3}
+ 5**

**(iii) The radius of a hydrogen atom is
2.5 × 10 ^{–11} m**

**Solution:**

**(i) 8 × 10 ^{4} + 7 × 10^{3} + 6 × 10^{2 } + 5 × 10^{1} + 2 × 1 + 4 × 10^{−2}
+ 7 × 10^{−4}**

(8 × 10^{4}) + (7 × 10^{3}) + (6
× 10^{2})^{ } + (5 × 10^{1})
+ (2 × 1) + (4 × 10^{−2}) + (7 × 10^{−4})^{}

= (8 × 10000) + (7 × 1000) + (6 × 100) + (5 × 10) + (2 × 1) + (4
× 1/100) + (7 × 1/10000)

= 80000 + 7000 + 600 + 50 + 2 + (4/100) + (7/10000)

= 87652.0407

**(ii) 5 × 10 ^{3} + 5 × 10^{1} + 5 × 10^{−1}
+ 5 × 10^{−3}**

**Solution:**

5 × 10^{3} + 5 × 10^{1} + 5 × 10^{−1} +
5 × 10^{−3}

= 5 × 1000 + 5 × 10 + 5 × 1/10 + 5 × 1/1000

= 5000 + 50 + 5/10 + 5/1000 = 5050.505

**(iii) The radius of a hydrogen atom is 2.5 × 10 ^{−11} m.**

**Solution:**

Radius of a hydrogen atom = 2.5 × 10^{−11} m

= 2.5 × 1/10^{11} m = 2.5/10^{11}m =
0.000000000025 m

** **

**10. Write the following numbers in scientific
notation:**

**(i) 467800000000 (ii) 0.000001972 (iii)
1642.398**

**(iv) Earth’s volume is about 1,083,000,000,000
cubic kilometres**

**(v) If you fill a bucket with dirt, the
portion of the whole Earth that is in the bucket will be 0.0000000000000000000000016
kg**

**Solution:**

** **

**Objective
Type Questions**

**11. By what number should ****(−****4****) ^{−}**

(A) 2/3

(B) −2/5

(C) 5/2

(D) −5/2

**[Answer: (B) −2/5]**

**Solution:**

(−4)^{−1} = (−1/4)^{1} = −1/4

10^{−1} = (1/10)^{1} = 1/10

(−1/4) × *x* = 1/10

*x
*= −4 / 10 = −2 / 5

**12. (−2) ^{−3} × (−2 )^{−2}
= ____________.**

(A) −1/32

(B) 1/32

(C) 32

(D) –32

**[Answer: (A) −1/32]**

**13. Which is not correct?**

**[Answer: (D) –(1/4) ^{2} = 16^{−1}]**

**Solution:**

(−2) – 3*x*( − 2) – 2 = (−2) – 3 – 2 = (−2)
−5 (−1/2)5 = −1/32

**14. If 10^{x}/10^{-3} =10^{9} ,then x is ____________.**

(A) 4

(B) 5

(C) 6

(D) 7

**[Answer: (C) 6]**

**15. 0.0000000002020 in scientific form
is ____________.**

(A) 2.02×10^{9}

(B) 2.02×10^{−}^{9}

(C) 2.02×10^{−}^{8}

(D) 2.02×10^{−}^{10}

[Answer: (D) 2.02 × l0−10]^{}

**Solution:**

^{ }

**Answer**

**Exercise 1.6 **

**1. (i) 1 (ii) 1 (iii) 20 ^{−3}
(iv) −1/128 (v) –243 **

**2. (i) True (ii) False
(iii) False (iv) True (v) False**

**3. (i) 1/8 (ii) 32
(iii) – 216/125 (iv) 16 (v) 6 **

**4. (i) 64/15625 (ii) 4/5
(iii) 1024 **

**5. (i) 21/2 (ii) 5 (iii)
1**

**6. (i) 1 (ii) 3/2
(iii) 72 **

**7. (i) x = 5
(ii) x = 6**

**8. (i) 6 × 10 ^{3}
+ 5 × 10^{1} + 4 × 10^{0} + 3 × 10^{‒1} + 2 × 10^{‒2}
+ 1 × 10^{-3} **

**(ii) 8 × 10 ^{2}
+ 9 × 10^{1} + 7 × 10^{0} + 1 × 10^{‒1} + 4 × 10^{‒2}**

**9. (i) 87652.0407 (ii)
5050.505 (iii) 0.000000000025**

**10. (i) 4.678 × 10 ^{11
}(ii) 1.972 × 10^{‒6} (iii) 1.642398 ×10^{3 }(iv) 1.083 ×10^{12}
cu. km (v) 1.6 ×10^{‒24}**

**11. (B) -2/5 **

**12. (A) -1/32**

**13. (D) – (1/4) ^{2}
= 16^{-1} **

**14. (C) 6 **

**15. (D) 2.02×10 ^{-10}**

^{}

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8th Maths : Chapter 1 : Numbers : Exercise 1.6 (Exponents and Powers) | Questions with Answers, Solution | Numbers | Chapter 1 | 8th Maths

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