Class 7 Maths Chapter 10 Question Answers - Exponents and Powers

Q1: Express the following numbers using scientific notation:
(A) 1,234.56
(B) 0.0876
Ans:
To express numbers in scientific notation, we write them in the form:
a × 10^b
where a is a number between 1 and 10 (excluding 10) and b is an integer representing the exponent of 10. Here's how we can express the given numbers in scientific notation:
(A) 1,234.56: 
1,234.56 = 1.23456 × 10³
(B) 0.0876: 
0.0876 = 8.76×10⁻²
Note that in scientific notation, the decimal point is moved to the right for positive exponents (b > 0) and to the left for negative exponents (b < 0) to get a number between 1 and 10. The exponent indicates how many places the decimal point was moved.

Q2: Simplify:
10⁻¹ × 10² × 10⁻³ × 10⁴ × 10⁻⁵ × 10⁶
Ans:
Given
10⁻¹ ×10² ×10⁻³ ×10⁴ ×10⁻⁵ ×10⁶
We know that x^m × xⁿ = x^(m+n) so simplifying using this as below|
= 10^(2+4+6) ×10 ^{(−1−3−5)}
= 10¹² ×10⁻⁹
= 10⁽¹²⁻⁹⁾
= 10³

Q3: Express the following information in the standard form:
Size of the bacteria is 0.0000004 m
Ans:
Size of the bacteria is 0.0000004 m
In standard form,
Size of the bacteria is 4 × 10⁻⁷  m

Q4: Convert 62000+39000 to scietific form.
Ans:
On adding, we get
62000 + 39000 = 101000
Therefore, 101000 = 1.01 × 10⁵

Q5: Prove that.
Ans:
Consider,


 
Multiply and divide each terms in the denominator by the missing terms. i.e., c,a,b respectively

= abc
Hence proved.

Q6:  Solve: .
Ans:
We know,

= 0.65934 x 10⁶
= 6.5934 x 10⁷

Q7: Convert into power notation: 156
Ans:
Given number is 156
156 = 2 × 2 × 3 × 13
156 = 2²  × 3 × 13

Q8: Simplify: .
Ans:

Formulae:


Q9: Evaluate .
Ans:


Q10: If the radian measures of two angles of a triangle are as given below . find the radian measure and the degree measure of the third angle then. 3π/5, 4π/15.
Ans:
We have two angles of the triangle

Let the third angle be x
So,
In Angle  x = 2 x 180 / 15 = 24^o

Q11: The number of zeroes at the end of the sum 101¹¹ −1.
Ans:
101 × 101 = 10201
(101)² = 1030301
(101)⁴  = 104060401
there is one pattern that in all the powers of 101 last three digits
M01 where M is a non zero digit.
So at the end of (101)ⁿ −1
last three digits are M00
then there are 2 zeroes at the end.

Q12: Express the following in exponential form:
(i) 6 × 6 × 6 × 6
(ii) t × t
(iii) b × b × b × b
(iv) 5 × 5 × 7 × 7 × 7
(v) 2 × 2 × a × a
(vi) a × a × a × c × c × c × c × d
Ans:
Now,
(i) 6 × 6 × 6 × 6 = 64 .  
(ii) t × t  = t² .
(iii) b × b × b × b = b4
(iv) 5 × 5 × 7 × 7 × 7 = 5² × 7³
(v) 2 × 2  × a × a = 2² × a²     
(vi) a × a × a  × c × c × c × c × d = a³ × c⁴ × d

Q13: Simplify: .
Ans:

= 2/3⁴
= 2/81
Hence, the answer is 2/81.

Q14: Evaluate .
Ans:

= 3⁸ x 2⁰
= 3⁸
= 6561

Q15: The population of India is 125,00000000. Write in exponential form.
Ans:
125,00000000 = 125 × 100000000
= 125 × 10⁸
= 1.25 × 100 × 10⁸
= 1.25 × 10²⁺⁸
= 1.25 × 10¹⁰