Samacheer Kalvi 8th Maths Guide Chapter 1 Numbers Ex 1.6

Tamilnadu State Board New Syllabus Samacheer Kalvi 8th Maths Guide Pdf Chapter 1 Numbers Ex 1.6 Textbook Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 8th Maths Solutions Chapter 1 Numbers Ex 1.6

Question 1.
Fill in the blanks.
(i) (-1)^{even integer} is __________ .
Answer:
1

(ii) For a ≠ 0, a⁰ is __________ .
Answer:
1

(iii) 4^-3 × 5^-3 = __________ .
Answer:
20^-3

(iv) (-2)^-7 is = __________ .
Answer:
\(\frac{-1}{128}\)

(v) \(\left(-\frac{1}{3}\right)^{-5}\) = _________ .
Answer:
– 243

Question 2.
Say True or False:
(i) If 8^x = \(\frac { 1 }{ 64 }\), the value of x is -2.
Answer:
True

(ii) The simplified form of \((256)^{\frac{-1}{4}} \times 4^{2}\) is \(\frac{1}{4}\).
Answer:
True

(iii) Using the power rule, \(\left(3^{7}\right)^{-2}\) = 3⁵
Answer:
True

(iv) The standard form of 2 × 10^-4 is 0.0002.
Answer:
False

(v) The scientific form of 123.456 is 1.23456 × 10^-2.
Answer:
True

Question 3.
Evaluate
(i) \(\left(\frac{1}{2}\right)^{3}\)
(ii) \(\left(\frac{1}{2}\right)^{-5}\)
(iii) \(\left(\frac{-5}{6}\right)^{-3}\)
(iv) (2^-5 × 2⁷) ÷ 2^-2
(v) (2^-1 × 3^-1) ÷ 6^-2
Answer:
(i) \(\left(\frac{1}{2}\right)^{3}\)

(ii) \(\left(\frac{1}{2}\right)^{-5}\)

(iii) \(\left(\frac{-5}{6}\right)^{-3}\)

(iv) (2^-5 × 2⁷) ÷ 2^-2
(2^-5 × 2⁷) ÷ 2^-2 = (2^{-5 + 7}) ÷ 2^-2
= 2² ÷ 2^-2
= 2²⁺²
= 2⁴
= 16

(v) (2^-1 × 3^-1) ÷ 6^-2
(2^-1 × 3^-1) ÷ 6^-2 = (2 × 3)^-1 ÷ 6^-2
= (6^-1) ÷ 6^-2
= 6^(-1)-(-2)
= 6¹
= 6

Question 4.
Evaluate
(i) \(\left(\frac{2}{5}\right)^{4} \times\left(\frac{5}{2}\right)^{-2}\)
(ii) \(\left(\frac{4}{5}\right)^{-2} \div\left(\frac{4}{5}\right)^{-3}\)
(iii) \(2^{7} \times\left(\frac{1}{2}\right)^{-3}\)
Answer:
(i) \(\left(\frac{2}{5}\right)^{4} \times\left(\frac{5}{2}\right)^{-2}\)

(ii) \(\left(\frac{4}{5}\right)^{-2} \div\left(\frac{4}{5}\right)^{-3}\)

(iii) \(2^{7} \times\left(\frac{1}{2}\right)^{-3}\)
= 2⁷ × 2³
= 2^{7 + 3}
= 2¹⁰

Question 5.
Evaluate:
(i) (5⁰ + 6^-1) × 3²
(ii) (2^-1 + 3^-1) ÷ 6^-1
(iii) (3^-1 + 4^-2 + 5^-3)⁰
Answer:
(i) (5⁰ + 6^-1) × 3²

(ii) (2^-1 + 3^-1) ÷ 6^-1
Answer:
(2^-1 + 3^-1) ÷ 6^-1 = \(\left(\frac{1}{2}+\frac{1}{3}\right)\) + 6^-1
= \(\left(\frac{3+2}{6}\right)\) + 6^-1 = \(\left(\frac{5}{6}\right)\) + 6^-1 = \(\frac{5}{6}\) × 6 = 5

(iii) (3^-1 + 4^-2 + 5^-3)⁰
Answer:
(3^-1 + 4^-2 + 5^-3)⁰ = 1
[∵ a⁰ = 1 where a ≠ 0]

Question 6.
Simplify
(i) (3²)³ × (2 × 3⁵)^-2 × (18)²
(ii) \(\frac{9^{2} \times 7^{3} \times 2^{5}}{84^{3}}\)
(iii) \(\frac{2^{8} \times 2187}{3^{5} \times 3^{2}}\)
Answer:
(i) (3²)³ × (2 × 3⁵)^-2 × (18)²

(ii) \(\frac{9^{2} \times 7^{3} \times 2^{5}}{84^{3}}\)

(iii) \(\frac{2^{8} \times 2187}{3^{5} \times 3^{2}}\)


= 2^8-5 × 3^7-5
= 2³ × 3²
= 8 × 9
= 72

Question 7.
Solve for x:
(i) \(\frac{2^{2 x-1}}{2^{x+2}}\) = 4
(ii) \(\frac{5^{5} \times 5^{-4} \times 5^{x}}{5^{12}}\) = 5^-5
Answer:
(i) \(\frac{2^{2 x-1}}{2^{x+2}}\) = 4
2^{2x – 1 – (x + 2)} = 2²
2^{2x – 1 – x – 2)} = 2²
2^{2x – 3} = 2²
Equating the powers of the same base 2.
x – 3 = 2
x – 3 + 3 = 2 + 3
x = 5

(ii) \(\frac{5^{5} \times 5^{-4} \times 5^{x}}{5^{12}}\) = 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

Question 8.
Expand using exponents:
(i) 6054.321
(ii) 897.14
Answer:
(i) 6054.321
6054.321 = (6 × 1000) + (0 × 100) + (5 × 10) + (4 × 10⁰) + \(\frac{3}{10}+\frac{2}{100}+\frac{1}{1000}\)
= (6 × 10³) + (5 × 10¹) + (4 × 10⁰) + \(\frac{3}{10}+\frac{2}{100}+\frac{1}{1000}\)
= (6 × 10³) + (5 × 10¹) + (4 × 10⁰) + (3 × 10^-1) + (2 × 10^-2) + (1 × 10^-3)

(ii) 897.14
= (8 × 100) + (9 × 10) + (7 × 10⁰) + \(\frac{1}{10}+\frac{4}{100}\)
= (8 × 1o²) +( 9 × 10¹) + (7 × 10⁰) + \(\left(1 \times \frac{1}{10}\right)+\left(4 \times \frac{1}{100}\right)\)
= (8 × 10³) + (9 × 10³) + (7 × 10⁰) + (1 × 10^-1) + (4 × 10^-2)

Question 9.
Find the number is standard form:
(i) 8 × 10⁴ + 7 × 10³ + 6 × 10² + 5 × 10¹ + 2 × 1 + 4 × 10^-2 + 7 × 10^-4
(ii) 5 × 10³ + 5 × 10¹ + 5 × 10^-1 + 5 × 10^-3
(iii) The radius of a hydrogen atom is 2.5 × 10^-11 m
Answer:
(i) 8 × 10⁴ + 7 × 10³ + 6 × 10² + 5 × 10¹ + 2 × 1 + 4 × 10^-2 + 7 × 10^-4
= 8 × 10⁴ + 7 × 10³ + 6 × 10² + 5 × 10¹ + 2 × 1 + 4 × 10^-2 + 7 × 10^-4
= 8 × 10000 + 7 × 1000 + 6 × 100 + 5 × 10 + 2 × 1 + 4 × \(\frac{1}{100}\) + 7 × \(\frac{1}{10000}\)
= 80000 + 7000 + 600 + 50 + 2 + \(\frac{4}{100}\) + \(\frac{7}{10000}\)
= 87652.0407

(ii) 5 × 10³ + 5 × 10¹ + 5 × 10^-1 + 5 × 10^-3
= 5 × 10³ + 5 × 10¹ + 5 × 10^-1 + 5 × 10^-3
= 5 × 1000 + 5 × 10 + 5 × \(\frac{1}{10}\) + 5 × \(\frac{1}{1000}\)
= 5000 + 50 + \(\frac{5}{10}+\frac{5}{1000}\) = 5050.505

(iii) The radius of a hydrogen atom is 2.5 10^-11 m
Radiys of a hydrogen atom = 2.5 × 10^-11 m
= \(2.5 \times \frac{1}{10^{11}} \mathrm{m}=\frac{2.5}{10^{11}} \mathrm{m}\)
= 0.000000000025 m

Question 10.
Write the following numbers in scientific notation:
(i) 467800000000
Answer:
467800000000 = 4.678 × 10¹¹

(ii) 0.000001972
Answer:
0.000001972 = 1.972 × 10^-6

(iii) 1642.398
Answer:
1642.398 = 1.642398 × 10³

(iv) Earth’s volume is about 1,083,000,000,000 cubic kilometres
Answer:
1,083,000,000,000
Earth’s volume = 1.083 110 × 10² cubic kilometres

(v) If you fill a bucket with dirt, the portion of the whole Earth that is in the bucket will be 0.00000000000000000000000 16 kg
Answer:
Portion of earth in the bucket = 0.00000000000000000000000 16 kg
= 1.6 10 × 10²⁴ kg.

Objective Type Questions

Question 11.
By what number should (-4)^-1 be multiplied so that the product becomes 10^-1?
(A) \(\frac{2}{3}\)
(B) \(\frac{-2}{5}\)
(C) \(\frac{5}{2}\)
(D) \(\frac{-5}{2}\)
Answer:
(B) \(\frac{-2}{5}\)
Hint:
(-4)^-1 = \(\left(-\frac{1}{4}\right)^{1}=\frac{-1}{4}\)

Question 12.
(-2)^-3 × (-2)^-2 = ___________.
(A) \(\frac{-1}{32}\)
(B) \(\frac{1}{32}\)
(C) 32
(D) -32
Answer:
(A) \(\frac{-1}{32}\)

Question 13.
Which is not correct?
(A) \(\left(\frac{-1}{4}\right)^{2}\) = 4^-2
(B) \(\left(\frac{-1}{4}\right)^{2}=\left(\frac{1}{2}\right)^{4}\)
(C) \(\left(\frac{-1}{4}\right)^{2}\) = 16^-1
(D) \(-\left(\frac{1}{4}\right)^{2}\) = 16^-1
Answer:
\(-\left(\frac{1}{4}\right)^{2}\) = 16^-1
Hint:
(-2) – 3 x (- 2) – 2 = (-2) – 3 – 2 = (-2) – 5 (\(-\frac { 1 }{ 2 }\))5 = \(-\frac { 1 }{ 32 }\)

Question 14.
If \(\frac{10^{x}}{10^{-3}}\) = 10⁹, then x is ___________ .
(A) 4
(B) 5
(C) 6
(D) 7

Question 15.
0.0000000002020 in scientific form is __________ .
(A) 2.02 × 10⁹
(B) 2.02 × 10^-9
(C) 2.02 × 10^-8
(D) 2.02 × 10^-10
Answer:
(D) 2.02 × 10^-10
Hint:
0.0000000002020