Powers
Class 8th Mathematics RD Sharma Solution
- Express each of the following as a rational number of the form q/p , where p and…
- (i) 3^-1 + 4^-1 (ii) (3^0 + 4^-1) x 2^2 (iii) (3^-1 + 4^-1 + 5^-1)^0 (iv)…
- (i) (1/2)^-1 + (1/3)^-1 + (1/4)^-1 (ii) (1/2)^-2 + (1/3)^-2 + (1/4)^-2 (iii)…
- (i) (4^-1 x 3^-1)^2 (ii) (5^-1 / 6^-1)^3 (iii) (2^-1 x 4^-1) / 2^-2 (iv) (3^-1 x…
- (i) (3^2 + 2^2) x (1/2)^3 (ii) (3^2 + 2^2) x (2/3)^-3 (iii) [(1/3)^-3 -…
- By what number should 5-1 be multiplied so that the product many be equal to…
- By what number should (1/2)^-1 be multiplied so that the product may be equal to…
- By what number should (-15)-1 be divided so that the quotient may be equal to…
- Write each of the following in exponential form: (i) (3/2)^-1 x (3/2)^-1 x…
- Evaluate: (i) 5-2 (ii) (-3)-2 (iii) (1/3)^-4 (iv) (-1/2)^-1
- Express each of the following as a rational number in the form p/q : (i) 6-1…
- (i) 4^-1 x 3^-1^2 (ii) 5^-1 / 6^-1^3 (iii) (2^-1 + 3^-1)^-1 (iv) 3^-1 x 4^-1^-1…
- Express each of the following rational numbers with a negative exponent: (i)…
- Express each of the following rational numbers with a positive exponent: (i)…
- (i) (1/3)^-3 - (1/2)^-3 / (1/4)^-3 (ii) (3^2 - 2^2) x (2/3)^-3 (iii) (1/2)^-1 x…
- By what number should 5-1 be multiplied so that the product may be equal to…
- By what number should (1/2)^-1 be multiplied so that the product may be equal to…
- By what number should (-15)-1 be divided so that the quotient may be equal to…
- By what number should (5/3)^-2 be multiplied so that the product may be…
- Find x, if (i) (1/4)^-4 x (1/4)^-8 x (1/4)^-4x (ii) (-1/2)^-19 / (-1/2)^8 =…
- If x= (3/2)^2 x (2/3)^-4 , find the value of x-2.
- If x = (4/5)^-2 / (1/4)^2 , find the value of x-1.
- Find the value of x for which 5^2x / 5^-3 = 5^5 .
Exercise 2.1
Question 1.Express each of the following as a rational number of the form
, where p and q are integers and q
0:
(i) 2-3
(ii) (-4)-2
(iii) 
(iv) 
(v) 
Answer:
(i) 2-3 = 
= 
[Using a-n = 
]
(ii) (-4)-2 = 
= 
= 
[Using a-n = ]
(iii) 
= 
[Using 
n]
(iv) 
= 
= 
= 
[Using 
-n = ]
(v) 
= 
= 
[Using -n = 
]
Question 2.
Find the values of each of the following:
(i) 
(ii) 
(iii)
(iv) 
Answer:
(i) 
⇒ 
(LCM of 3 and 4 is 12) [Using -n = ]
(ii) 
⇒ 
(LCM of 1 and 4 is 4) [Using -n = 
]
(iii)
⇒ 
[Using 
]
We know that any number to power zero is always equal to 1.
(iv) 
⇒ 
[Using -n = ]
Question 3.
Find the values of each of the following:
(i) 
(ii) 
(iii)
(iv) 
Answer:
(i) 
⇒ 
[Using -n = 
]
(ii) 
⇒ 
[Using -n = ]
(iii)
⇒ 
[Using -n = 
]
[Using 
]
⇒ 
(iv) 
⇒ 
[Using -n = ]
[Using 
]
⇒ 
= 
Question 4.
Simplify:
(i) 
(ii) 
(iii) 
(iv) 
Answer:
(i) 
⇒ 
[Using -n = 
]
(ii) 
⇒ 
[Using -n = 
]
[Using 
]
(iii)
⇒ 
[Using -n = ]
[Using 
]
(iv) 
⇒ 
[Using -n = ]
Question 5.
Simplify:
(i) 
(ii) 
(iii) 
(iv) 
Answer:
(i) 
⇒ 
[Using 
]
(ii) 
⇒ 
[Using -n = 
]
(iii) 
⇒ 
[Using -n = ]
⇒ 
[Using ]
⇒ 
[Using
]
(iv) 
⇒ 
[Using ]
⇒ 
[Using ]
Question 6.
By what number should 5-1 be multiplied so that the product many be equal to (-7)-1?
Answer:
Let the number 5-1 shlould be multiplied by 
According to the question:
5-1 = (-7)-1
[Using -n = ]
Therefore 5-1 should be multiplied by 
Question 7.
By what number should
be multiplied so that the product may be equal to 
?
Answer:
Let the number 
shlould be multiplied by
According to the question:
= 
[Using -n = ]
Therefore should be multiplied by 
Question 8.
By what number should (-15)-1 be divided so that the quotient may be equal to (-5)-1?
Answer:
Let the number (-15)-1 shlould be devided by 
According to the question:
(-15)-1 ÷
= (-5)-1
[Using 
-n = 
]
[using 
]
Therefore (-15)-1 should be devided by 
Exercise 2.2
Question 1.Write each of the following in exponential form:
(i) 
(ii) 
Answer:
(i) 
⇒ 
[Using -n = ]
(ii) 
⇒
[Using -n = ]
Question 2.
Evaluate:
(i) 5-2
(ii) (-3)-2
(iii) 
(iv)
Answer:
(i) 5-2
⇒ 
[Using -n = ]
(ii) (-3)-2
⇒ 
[Using -n = ]
(iii) 
⇒ 
[Using -n = ]
(iv)
⇒ 
[Using -n = ]
Question 3.
Express each of the following as a rational number in the form
:
(i) 6-1
(ii) (-7)-1
(iii) 
(iv) 
(v) 
Answer:
(i) 6-1
⇒ 6-1 = 
[Using -n = ]
(ii) (-7)-1
⇒ (-7)-1 = 
[Using -n = ]
(iii) 
⇒ = 4 [Using -n = ]
(iv) 
⇒ = 
[Using -n = ]
(v) 
⇒ = 
[Using -n = ]
Question 4.
Simplify:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
Answer:
(i) 
⇒ 
[Using -n = ]
(ii) 
⇒ 
[Using -n = ]
⇒ ( 
[Using ]
⇒ 
[Using ]
(iii) 
⇒ 
[Using -n = ]
⇒ 
[Using -n = ]
(iv) 
⇒ 
[Using -n = ]
⇒ 
[Using -n = ]
(v) 
⇒ 
[Using -n = ]
⇒ 
[Using ]
⇒ 
Question 5.
Express each of the following rational numbers with a negative exponent:
(i) 
(ii) 35
(iii) 
(iv) 
(v) 
Answer:
(i) 
⇒ 
[Using 
-n]
(ii) 35
⇒ 35
[Using -n]
(iii) 
⇒ = 
[Using 
]
(iv) 
⇒ = 
[Using (aⁿ)ᵐ = aᵐⁿ ]
(v) 
⇒ = 
[Using (aⁿ)ᵐ = aᵐⁿ ]
Question 6.
Express each of the following rational numbers with a positive exponent:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
Answer:
(i) 
⇒ 
= 
[Using ]
(ii) 
⇒ = 
[Using ]
(iii) 
⇒ = 43–9 = 4–6 [Using (aⁿ× aᵐ = aᵐ⁺ⁿ ]
⇒ 4–6 = 
[Using -n]
(iv) 
⇒ = 
[Using (aⁿ)ᵐ = aᵐⁿ ]
(v) 
⇒ = 
[Using (aⁿ)ᵐ = aᵐⁿ and -n ]
Question 7.
Simplify:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
Answer:
(i) 
⇒
⇒ 
[Using -n ]
⇒ (27-8) ÷ 43
⇒ 19 ÷ 43
⇒ [Using ]
(ii) 
⇒
⇒ 
[Using -n ]
⇒ 
⇒ 
[Using ]
⇒ 
(iii) 
⇒
⇒ 
[Using -n ]
⇒ 
⇒
(iv) 
⇒
⇒ 
[Using (aⁿ)ᵐ = aᵐⁿ]
⇒ 
[Using ]
(v) 
Question 8.
By what number should 5-1 be multiplied so that the product may be equal to (-7)-1 ?
Answer:
Let the number 5-1 shlould be multiplied by
According to the question:
5-1 = (-7)-1
[Using -n = ]
[Using ]
Therefore 5-1 should be multiplied by
Question 9.
By what number should
be multiplied so that the product may be equal to 
?
Answer:
Let the number shlould be multiplied by
According to the question:
=
[Using -n = ]
[Using ]
Therefore 
should be multiplied by
Question 10.
By what number should (-15)-1 be divided so that the quotient may be equal to (-5)-1 ?
Answer:
Let the number (-15)-1 shlould be devided by
According to the question:
(-15)-1 ÷
= (-5)-1
[Using -n = ]
[Using ]
Therefore (-15)-1 should be devided by 
Question 11.
By what number should
be multiplied so that the product may be
?
Answer:
Let the number 
shlould be multiplied by
According to the question:
= 
[Using -n = ]
[Using ]
Therefore should be multiplied by 
Question 12.
Find x, if
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
Answer:
(i) 
[Using n × m = ᵐ⁺n]
Equating coefficients when bases are equal.
(ii) 
[Using n ÷ m = ᵐ–n]
Equating coefficients when bases are equal.
(iii) 
[Using n × m = ᵐ⁺n]
Equating coefficients when bases are equal.
(iv) 
[Using n × m = ᵐ⁺n]
Equating coefficients when bases are equal.
(v) 
[Using n ÷ m = ᵐ–n]
Equating coefficients when bases are equal.
(vi) 
[Using n × m = ᵐ⁺n]
Equating coefficients when bases are equal.
Question 13.
If x=
, find the value of x-2.
Answer:
x =
[Using am × an = am+n ]
Question 14.
If x = 
, find the value of x-1.
Answer:
X = 
On using, [Using -n = ], we get,
[Using ]
x = 25
Question 15.
Find the value of x for which
.
Answer:
[Using n ÷ m = ᵐ–n]
Equating coefficients when bases are equal.
Exercise 2.3
Question 1.Express the following numbers in standard form:
(i) 6020000000000000
(ii) 0.00000000000942
(iii) 0.00000000085
(iv) 
(v) 
(vi) 0.00072984
(vii) 0.000437
(viii) 
Answer:
(i) 6020000000000000
To write in the standard form, count the number of digits leaving one digit from the left. The total number of digits so obtained becomes power of 10. Decimal comes after first left digit.
In this question total number of digits leaving one digit from left are 15.
Therefore the standard form is: 6.02× 1015
(ii) 0.00000000000942
To write in the standard form, count the number of digits leaving one digit from the left. The total number of digits so obtained becomes power of 10. Decimal comes after first left digit.
If the number has all the digits to the right of the decimal then powers will be negative. In this question total number of digits after decimal are 12.
Therefore the standard form is: 9.42× 10–12
(iii) 0.00000000085
To write in the standard form, count the number of digits leaving one digit from the left. The total number of digits so obtained becomes power of 10. Decimal comes after first left digit.
If the number has all the digits to the right of the decimal then powers will be negative. In this question total number of digits after decimal are 12.
Therefore the standard form is: 8.5× 10–10
(iv) 
To write in the standard form, count the number of digits leaving one digit from the left. The total number of digits so obtained becomes power of 10. Decimal comes after first left digit.
In this question total number of digits are 2.
Therefore the standard form is: 8.46× 109
To write in the standard form, count the number of digits leaving one digit from the left. The total number of digits so obtained becomes power of 10. Decimal comes after first left digit.
In this question total number of digits are 3.
Therefore the standard form is: 8.46× 10–1
To write in the standard form, count the number of digits leaving one digit from the left. The total number of digits so obtained becomes power of 10. Decimal comes after first left digit.
If the number has all the digits to the right of the decimal then powers will be negative. In this question total number of digits after decimal are 4.
Therefore the standard form is: 7.2984× 10–4
To write in the standard form, count the number of digits leaving one digit from the left. The total number of digits so obtained becomes power of 10. Decimal comes after first left digit.
If the number has all the digits to the right of the decimal then powers will be negative. In this question total number of digits after decimal are 4.
Therefore the standard form is: 4.37
To write in the standard form, Count the number of zeros of the divisor. This number of zeros becomes negative power of 10.
Therefore the standard form is: 4× 10–5
Question 2.
Write the following numbers in the usual form:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
Answer:
In cae of positive power of 10. The usual form of the number is written after multiplying the given numbers and then counts the numbers from the right and put decimal.
Step 1: Multipy the given numbers: 4.83× 10000000 = 4830000000
Step 2: Put decimal after two places from the right: 48300000.00
Step 3: Write the number after in the usual form: 48300000
In case of negative powers, decimal sifts to left equal to the power of 10.
Step 1: Here the power of 10 is negative 6.
Step 2: Therefore decimal will sift six places to the left. i.e: 0.00000302
In cae of positive power of 10. The usual form of the number is written after multiplying the given numbers and then counts the numbers from the right and put decimal.
Step 1: Multipy the given numbers: 4.5× 10000 = 450000
Step 2: Put decimal after one place from the right: 45000.0
Step 3: Write the number after in the usual form: 45000
In case of negative powers, decimal sifts to left equal to the power of 10.
Step 1: Here the power of 10 is negative 8.
Step 2: Therefore decimal will sift eight places to the left, and we write zeros before the number to make eight places . i.e: 0.00000003
In cae of positive power of 10. The usual form of the number is written after multiplying the given numbers and then counts the numbers from the right and put decimal.
Step 1: Multipy the given numbers: 1.0001× 1000000000 = 10001000000000
Step 2: Put decimal after four places from the right: 1000100000.0000
Step 3: Write the number after in the usual form: 1000100000
In cae of positive power of 10. The usual form of the number is written after multiplying the given numbers and then counts the numbers from the right and put decimal.
Step 1: Multipy the given numbers: 5.8× 100 = 5800
Step 2: Put decimal after one place from the right: 580.0
Step 3: Write the number after in the usual form: 580
In cae of positive power of 10. The usual form of the number is written after multiplying the given numbers and then counts the numbers from the right and put decimal.
Step 1: Multipy the given numbers: 3.61492 × 1000000 = 5800
Step 2: Put decimal after five places from the right: 3614920.00000
Step 3: Write the number after in the usual form: 3614920
In case of negative powers, decimal sifts to left equal to the power of 10.
Step 1: Here the power of 10 is negative 8.
Step 2: Therefore decimal will sift seven places to the left, and we write zeros before the number to make seven places . i.e: 0.000000325