Class 7 Exam > Class 7 Notes > Mathematics (Maths) Class 7 (Old NCERT) > RD Sharma Solutions: Exponents (Exercise 6.2)
Exponents (Exercise 6.2) RD Sharma Solutions | Mathematics (Maths) Class 7 (Old NCERT) PDF Download
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Page 1
Exercise 6.2 Page No: 6.28
1. Using laws of exponents, simplify and write the answer in exponential form
(i) 2
3
× 2
4
× 2
5
(ii) 5
12
÷ 5
3
(iii) (7
2
)
3
(iv) (3
2
)
5
÷ 3
4
(v) 3
7
× 2
7
(vi) (5
21
÷ 5
13
) × 5
7
Solution:
(i) Given 2
3
× 2
4
× 2
5
We know that first law of exponents states that a
m
× a
n
× a
p
= a
(m+n+p)
Therefore above equation can be written as 2
3
x 2
4
x 2
5
= 2
(3 + 4 + 5)
= 2
12
(ii) Given 5
12
÷ 5
3
According to the law of exponents we have a
m
÷ a
n
= a
m-n
Therefore given question can be written as 5
12
÷ 5
3
= 5
12 - 3
= 5
9
(iii) Given (7
2
)
3
According to the law of exponents we have (a
m
)
n
= a
mn
Therefore given question can be written as (7
2
)
3
= 7
6
(iv) Given (3
2
)
5
÷ 3
4
According to the law of exponents we have (a
m
)
n
= a
mn
Therefore (3
2
)
5
÷ 3
4
= 3
10
÷ 3
4
According to the law of exponents we have a
m
÷ a
n
= a
m-n
3
10
÷ 3
4
= 3
(10 - 4)
= 3
6
(v) Given 3
7
× 2
7
We know that law of exponents states that a
m
x b
m
= (a x b)
m
3
7
× 2
7
= (3 x 2)
7
= 6
7
(vi) Given (5
21
÷ 5
13
) × 5
7
According to the law of exponents we have a
m
÷ a
n
= a
m-n
= 5
(21 -13)
x 5
7
Page 2
Exercise 6.2 Page No: 6.28
1. Using laws of exponents, simplify and write the answer in exponential form
(i) 2
3
× 2
4
× 2
5
(ii) 5
12
÷ 5
3
(iii) (7
2
)
3
(iv) (3
2
)
5
÷ 3
4
(v) 3
7
× 2
7
(vi) (5
21
÷ 5
13
) × 5
7
Solution:
(i) Given 2
3
× 2
4
× 2
5
We know that first law of exponents states that a
m
× a
n
× a
p
= a
(m+n+p)
Therefore above equation can be written as 2
3
x 2
4
x 2
5
= 2
(3 + 4 + 5)
= 2
12
(ii) Given 5
12
÷ 5
3
According to the law of exponents we have a
m
÷ a
n
= a
m-n
Therefore given question can be written as 5
12
÷ 5
3
= 5
12 - 3
= 5
9
(iii) Given (7
2
)
3
According to the law of exponents we have (a
m
)
n
= a
mn
Therefore given question can be written as (7
2
)
3
= 7
6
(iv) Given (3
2
)
5
÷ 3
4
According to the law of exponents we have (a
m
)
n
= a
mn
Therefore (3
2
)
5
÷ 3
4
= 3
10
÷ 3
4
According to the law of exponents we have a
m
÷ a
n
= a
m-n
3
10
÷ 3
4
= 3
(10 - 4)
= 3
6
(v) Given 3
7
× 2
7
We know that law of exponents states that a
m
x b
m
= (a x b)
m
3
7
× 2
7
= (3 x 2)
7
= 6
7
(vi) Given (5
21
÷ 5
13
) × 5
7
According to the law of exponents we have a
m
÷ a
n
= a
m-n
= 5
(21 -13)
x 5
7
= 5
8
x 5
7
According to the law of exponents we have a
m
x a
n
= a
(m +n)
= 5
(8+7)
= 5
15
2. Simplify and express each of the following in exponential form:
(i) {(2
3
)
4
× 28} ÷ 2
12
(ii) (8
2
× 8
4
) ÷ 8
3
(iii) (5
7
/5
2
) × 5
3
(iv) (5
4
× x
10
y
5
)/ (5
4
× x
7
y
4
)
Solution:
(i) Given {(2
3
)
4
× 28} ÷ 2
12
{(2
3
)
4
x 2
8
} ÷ 2
12
= {2
12
x 2
8
} ÷ 2
12
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= 2
(12 + 8)
÷ 2
12
[According to the law of exponents we have a
m
x a
n
= a
(m +n)
]
= 2
20
÷ 2
12
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 2
(20 - 12)
=
2
8
(ii) Given (8
2
× 8
4
) ÷ 8
3
(8
2
× 8
4
) ÷ 8
3
[According to the law of exponents we have a
m
x a
n
= a
(m +n)
]
= 8
(2 + 4)
÷ 8
3
= 8
6
÷ 8
3
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 8
(6-3)
= 8
3
= (2
3
)
3
= 2
9
(iii) Given (5
7
/5
2
) × 5
3
= 5
(7-2)
x 5
3
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 5
5
x 5
3
[According to the law of exponents we have a
m
x a
n
= a
(m +n)
]
= 5
(5 + 3)
= 5
8
(iv) Given (5
4
× x
10
y
5
)/ (5
4
× x
7
y
4
)
= (5
4-4
× x
10-7
y
5-4
) [According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 5
0
x
3
y
1
[since 5
0
= 1]
= 1x
3
y
3. Simplify and express each of the following in exponential form:
(i) {(3
2
)
3
× 2
6
} × 5
6
(ii) (x/y)
12
× y
24
× (2
3
)
4
Page 3
Exercise 6.2 Page No: 6.28
1. Using laws of exponents, simplify and write the answer in exponential form
(i) 2
3
× 2
4
× 2
5
(ii) 5
12
÷ 5
3
(iii) (7
2
)
3
(iv) (3
2
)
5
÷ 3
4
(v) 3
7
× 2
7
(vi) (5
21
÷ 5
13
) × 5
7
Solution:
(i) Given 2
3
× 2
4
× 2
5
We know that first law of exponents states that a
m
× a
n
× a
p
= a
(m+n+p)
Therefore above equation can be written as 2
3
x 2
4
x 2
5
= 2
(3 + 4 + 5)
= 2
12
(ii) Given 5
12
÷ 5
3
According to the law of exponents we have a
m
÷ a
n
= a
m-n
Therefore given question can be written as 5
12
÷ 5
3
= 5
12 - 3
= 5
9
(iii) Given (7
2
)
3
According to the law of exponents we have (a
m
)
n
= a
mn
Therefore given question can be written as (7
2
)
3
= 7
6
(iv) Given (3
2
)
5
÷ 3
4
According to the law of exponents we have (a
m
)
n
= a
mn
Therefore (3
2
)
5
÷ 3
4
= 3
10
÷ 3
4
According to the law of exponents we have a
m
÷ a
n
= a
m-n
3
10
÷ 3
4
= 3
(10 - 4)
= 3
6
(v) Given 3
7
× 2
7
We know that law of exponents states that a
m
x b
m
= (a x b)
m
3
7
× 2
7
= (3 x 2)
7
= 6
7
(vi) Given (5
21
÷ 5
13
) × 5
7
According to the law of exponents we have a
m
÷ a
n
= a
m-n
= 5
(21 -13)
x 5
7
= 5
8
x 5
7
According to the law of exponents we have a
m
x a
n
= a
(m +n)
= 5
(8+7)
= 5
15
2. Simplify and express each of the following in exponential form:
(i) {(2
3
)
4
× 28} ÷ 2
12
(ii) (8
2
× 8
4
) ÷ 8
3
(iii) (5
7
/5
2
) × 5
3
(iv) (5
4
× x
10
y
5
)/ (5
4
× x
7
y
4
)
Solution:
(i) Given {(2
3
)
4
× 28} ÷ 2
12
{(2
3
)
4
x 2
8
} ÷ 2
12
= {2
12
x 2
8
} ÷ 2
12
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= 2
(12 + 8)
÷ 2
12
[According to the law of exponents we have a
m
x a
n
= a
(m +n)
]
= 2
20
÷ 2
12
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 2
(20 - 12)
=
2
8
(ii) Given (8
2
× 8
4
) ÷ 8
3
(8
2
× 8
4
) ÷ 8
3
[According to the law of exponents we have a
m
x a
n
= a
(m +n)
]
= 8
(2 + 4)
÷ 8
3
= 8
6
÷ 8
3
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 8
(6-3)
= 8
3
= (2
3
)
3
= 2
9
(iii) Given (5
7
/5
2
) × 5
3
= 5
(7-2)
x 5
3
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 5
5
x 5
3
[According to the law of exponents we have a
m
x a
n
= a
(m +n)
]
= 5
(5 + 3)
= 5
8
(iv) Given (5
4
× x
10
y
5
)/ (5
4
× x
7
y
4
)
= (5
4-4
× x
10-7
y
5-4
) [According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 5
0
x
3
y
1
[since 5
0
= 1]
= 1x
3
y
3. Simplify and express each of the following in exponential form:
(i) {(3
2
)
3
× 2
6
} × 5
6
(ii) (x/y)
12
× y
24
× (2
3
)
4
(iii)(5/2)
6
× (5/2)
2
(iv) (2/3)
5
× (3/5)
5
Solution:
(i) Given {(3
2
)
3
× 2
6
} × 5
6
= {3
6
× 2
6
} × 5
6
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= 6
6
× 5
6
[since law of exponents states that a
m
x b
m
= (a x b)
m
]
= 30
6
(ii) Given (x/y)
12
× y
24
× (2
3
)
4
= (x
12
/y
12
) × y
24
× 2
12
= x
12
× y
24-12
× 2
12
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= x
12
× y
12
× 2
12
= (2xy)
12
(iii) Given (5/2)
6
× (5/2)
2
= (5/2)
6+2
[According to the law of exponents we have a
m
x a
n
= a
(m +n)
]
= (5/2)
8
(iv) Given (2/3)
5
× (3/5)
5
= (2/5)
5
[since law of exponents states that a
m
x b
m
= (a x b)
m
]
4. Write 9 × 9 × 9 × 9 × 9 in exponential form with base 3.
Solution:
Given 9 × 9 × 9 × 9 × 9 = (9)
5
= (3
2
)
5
= 3
10
5. Simplify and write each of the following in exponential form:
(i) (25)
3
÷ 5
3
(ii) (81)
5
÷ (3
2
)
5
(iii) 9
8
× (x
2
)
5
/ (27)
4
× (x
3
)
2
(iv) 3
2
× 7
8
× 13
6
/ 21
2
× 91
3
Solution:
(i) Given (25)
3
÷ 5
3
= (5
2
)
3
÷ 5
3
[According to the law of exponents we have (a
m
)
n
= a
mn
]
Page 4
Exercise 6.2 Page No: 6.28
1. Using laws of exponents, simplify and write the answer in exponential form
(i) 2
3
× 2
4
× 2
5
(ii) 5
12
÷ 5
3
(iii) (7
2
)
3
(iv) (3
2
)
5
÷ 3
4
(v) 3
7
× 2
7
(vi) (5
21
÷ 5
13
) × 5
7
Solution:
(i) Given 2
3
× 2
4
× 2
5
We know that first law of exponents states that a
m
× a
n
× a
p
= a
(m+n+p)
Therefore above equation can be written as 2
3
x 2
4
x 2
5
= 2
(3 + 4 + 5)
= 2
12
(ii) Given 5
12
÷ 5
3
According to the law of exponents we have a
m
÷ a
n
= a
m-n
Therefore given question can be written as 5
12
÷ 5
3
= 5
12 - 3
= 5
9
(iii) Given (7
2
)
3
According to the law of exponents we have (a
m
)
n
= a
mn
Therefore given question can be written as (7
2
)
3
= 7
6
(iv) Given (3
2
)
5
÷ 3
4
According to the law of exponents we have (a
m
)
n
= a
mn
Therefore (3
2
)
5
÷ 3
4
= 3
10
÷ 3
4
According to the law of exponents we have a
m
÷ a
n
= a
m-n
3
10
÷ 3
4
= 3
(10 - 4)
= 3
6
(v) Given 3
7
× 2
7
We know that law of exponents states that a
m
x b
m
= (a x b)
m
3
7
× 2
7
= (3 x 2)
7
= 6
7
(vi) Given (5
21
÷ 5
13
) × 5
7
According to the law of exponents we have a
m
÷ a
n
= a
m-n
= 5
(21 -13)
x 5
7
= 5
8
x 5
7
According to the law of exponents we have a
m
x a
n
= a
(m +n)
= 5
(8+7)
= 5
15
2. Simplify and express each of the following in exponential form:
(i) {(2
3
)
4
× 28} ÷ 2
12
(ii) (8
2
× 8
4
) ÷ 8
3
(iii) (5
7
/5
2
) × 5
3
(iv) (5
4
× x
10
y
5
)/ (5
4
× x
7
y
4
)
Solution:
(i) Given {(2
3
)
4
× 28} ÷ 2
12
{(2
3
)
4
x 2
8
} ÷ 2
12
= {2
12
x 2
8
} ÷ 2
12
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= 2
(12 + 8)
÷ 2
12
[According to the law of exponents we have a
m
x a
n
= a
(m +n)
]
= 2
20
÷ 2
12
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 2
(20 - 12)
=
2
8
(ii) Given (8
2
× 8
4
) ÷ 8
3
(8
2
× 8
4
) ÷ 8
3
[According to the law of exponents we have a
m
x a
n
= a
(m +n)
]
= 8
(2 + 4)
÷ 8
3
= 8
6
÷ 8
3
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 8
(6-3)
= 8
3
= (2
3
)
3
= 2
9
(iii) Given (5
7
/5
2
) × 5
3
= 5
(7-2)
x 5
3
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 5
5
x 5
3
[According to the law of exponents we have a
m
x a
n
= a
(m +n)
]
= 5
(5 + 3)
= 5
8
(iv) Given (5
4
× x
10
y
5
)/ (5
4
× x
7
y
4
)
= (5
4-4
× x
10-7
y
5-4
) [According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 5
0
x
3
y
1
[since 5
0
= 1]
= 1x
3
y
3. Simplify and express each of the following in exponential form:
(i) {(3
2
)
3
× 2
6
} × 5
6
(ii) (x/y)
12
× y
24
× (2
3
)
4
(iii)(5/2)
6
× (5/2)
2
(iv) (2/3)
5
× (3/5)
5
Solution:
(i) Given {(3
2
)
3
× 2
6
} × 5
6
= {3
6
× 2
6
} × 5
6
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= 6
6
× 5
6
[since law of exponents states that a
m
x b
m
= (a x b)
m
]
= 30
6
(ii) Given (x/y)
12
× y
24
× (2
3
)
4
= (x
12
/y
12
) × y
24
× 2
12
= x
12
× y
24-12
× 2
12
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= x
12
× y
12
× 2
12
= (2xy)
12
(iii) Given (5/2)
6
× (5/2)
2
= (5/2)
6+2
[According to the law of exponents we have a
m
x a
n
= a
(m +n)
]
= (5/2)
8
(iv) Given (2/3)
5
× (3/5)
5
= (2/5)
5
[since law of exponents states that a
m
x b
m
= (a x b)
m
]
4. Write 9 × 9 × 9 × 9 × 9 in exponential form with base 3.
Solution:
Given 9 × 9 × 9 × 9 × 9 = (9)
5
= (3
2
)
5
= 3
10
5. Simplify and write each of the following in exponential form:
(i) (25)
3
÷ 5
3
(ii) (81)
5
÷ (3
2
)
5
(iii) 9
8
× (x
2
)
5
/ (27)
4
× (x
3
)
2
(iv) 3
2
× 7
8
× 13
6
/ 21
2
× 91
3
Solution:
(i) Given (25)
3
÷ 5
3
= (5
2
)
3
÷ 5
3
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= 5
6
÷
5
3
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 5
6 – 3
= 5
3
(ii) Given (81)
5
÷ (3
2
)
5
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= (81)
5
÷ 3
10
[81 = 3
4
]
= (3
4
)
5
÷ 3
10
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= 3
20
÷ 3
10
= 3
20-10
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 3
10
(iii) Given 9
8
× (x
2
)
5
/ (27)
4
× (x
3
)
2
= (3
2
)
8
× (x
2
)
5
/ (3
3
)
4
× (x
3
)
2
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= 3
16
× x
10
/3
12
× x
6
= 3
16-12
× x
10-6
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 3
4
× x
4
= (3x)
4
(iv) Given (3
2
× 7
8
× 13
6
)/ (21
2
× 91
3
)
= (3
2
× 7
2
7
6
× 13
6
)/(21
2
× 13
3
× 7
3
)[According to the law of exponents we have (a
m
)
n
= a
mn
]
= (21
2
× 7
6
× 13
6
)/(21
2
× 13
3
× 7
3
)
= (7
6
× 13
6
)/(13
3
× 7
3
)
= 91
6
/91
3
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 91
6-3
= 91
3
6. Simplify:
(i) (3
5
)
11
× (3
15
)
4
– (3
5
)
18
× (3
5
)
5
(ii) (16 × 2
n+1
– 4 × 2
n
)/(16 × 2
n+2
– 2 × 2
n+2
)
(iii) (10 × 5
n+1
+ 25 × 5
n
)/(3 × 5
n+2
+ 10 × 5
n+1
)
(iv) (16)
7
×(25)
5
× (81)
3
/(15)
7
×(24)
5
× (80)
3
Solution:
(i) Given (3
5
)
11
× (3
15
)
4
– (3
5
)
18
× (3
5
)
5
= (3)
55
× (3)
60
– (3)
90
× (3)
25
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= 3
55+60
– 3
90+25
= 3
115
- 3
115
Page 5
Exercise 6.2 Page No: 6.28
1. Using laws of exponents, simplify and write the answer in exponential form
(i) 2
3
× 2
4
× 2
5
(ii) 5
12
÷ 5
3
(iii) (7
2
)
3
(iv) (3
2
)
5
÷ 3
4
(v) 3
7
× 2
7
(vi) (5
21
÷ 5
13
) × 5
7
Solution:
(i) Given 2
3
× 2
4
× 2
5
We know that first law of exponents states that a
m
× a
n
× a
p
= a
(m+n+p)
Therefore above equation can be written as 2
3
x 2
4
x 2
5
= 2
(3 + 4 + 5)
= 2
12
(ii) Given 5
12
÷ 5
3
According to the law of exponents we have a
m
÷ a
n
= a
m-n
Therefore given question can be written as 5
12
÷ 5
3
= 5
12 - 3
= 5
9
(iii) Given (7
2
)
3
According to the law of exponents we have (a
m
)
n
= a
mn
Therefore given question can be written as (7
2
)
3
= 7
6
(iv) Given (3
2
)
5
÷ 3
4
According to the law of exponents we have (a
m
)
n
= a
mn
Therefore (3
2
)
5
÷ 3
4
= 3
10
÷ 3
4
According to the law of exponents we have a
m
÷ a
n
= a
m-n
3
10
÷ 3
4
= 3
(10 - 4)
= 3
6
(v) Given 3
7
× 2
7
We know that law of exponents states that a
m
x b
m
= (a x b)
m
3
7
× 2
7
= (3 x 2)
7
= 6
7
(vi) Given (5
21
÷ 5
13
) × 5
7
According to the law of exponents we have a
m
÷ a
n
= a
m-n
= 5
(21 -13)
x 5
7
= 5
8
x 5
7
According to the law of exponents we have a
m
x a
n
= a
(m +n)
= 5
(8+7)
= 5
15
2. Simplify and express each of the following in exponential form:
(i) {(2
3
)
4
× 28} ÷ 2
12
(ii) (8
2
× 8
4
) ÷ 8
3
(iii) (5
7
/5
2
) × 5
3
(iv) (5
4
× x
10
y
5
)/ (5
4
× x
7
y
4
)
Solution:
(i) Given {(2
3
)
4
× 28} ÷ 2
12
{(2
3
)
4
x 2
8
} ÷ 2
12
= {2
12
x 2
8
} ÷ 2
12
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= 2
(12 + 8)
÷ 2
12
[According to the law of exponents we have a
m
x a
n
= a
(m +n)
]
= 2
20
÷ 2
12
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 2
(20 - 12)
=
2
8
(ii) Given (8
2
× 8
4
) ÷ 8
3
(8
2
× 8
4
) ÷ 8
3
[According to the law of exponents we have a
m
x a
n
= a
(m +n)
]
= 8
(2 + 4)
÷ 8
3
= 8
6
÷ 8
3
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 8
(6-3)
= 8
3
= (2
3
)
3
= 2
9
(iii) Given (5
7
/5
2
) × 5
3
= 5
(7-2)
x 5
3
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 5
5
x 5
3
[According to the law of exponents we have a
m
x a
n
= a
(m +n)
]
= 5
(5 + 3)
= 5
8
(iv) Given (5
4
× x
10
y
5
)/ (5
4
× x
7
y
4
)
= (5
4-4
× x
10-7
y
5-4
) [According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 5
0
x
3
y
1
[since 5
0
= 1]
= 1x
3
y
3. Simplify and express each of the following in exponential form:
(i) {(3
2
)
3
× 2
6
} × 5
6
(ii) (x/y)
12
× y
24
× (2
3
)
4
(iii)(5/2)
6
× (5/2)
2
(iv) (2/3)
5
× (3/5)
5
Solution:
(i) Given {(3
2
)
3
× 2
6
} × 5
6
= {3
6
× 2
6
} × 5
6
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= 6
6
× 5
6
[since law of exponents states that a
m
x b
m
= (a x b)
m
]
= 30
6
(ii) Given (x/y)
12
× y
24
× (2
3
)
4
= (x
12
/y
12
) × y
24
× 2
12
= x
12
× y
24-12
× 2
12
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= x
12
× y
12
× 2
12
= (2xy)
12
(iii) Given (5/2)
6
× (5/2)
2
= (5/2)
6+2
[According to the law of exponents we have a
m
x a
n
= a
(m +n)
]
= (5/2)
8
(iv) Given (2/3)
5
× (3/5)
5
= (2/5)
5
[since law of exponents states that a
m
x b
m
= (a x b)
m
]
4. Write 9 × 9 × 9 × 9 × 9 in exponential form with base 3.
Solution:
Given 9 × 9 × 9 × 9 × 9 = (9)
5
= (3
2
)
5
= 3
10
5. Simplify and write each of the following in exponential form:
(i) (25)
3
÷ 5
3
(ii) (81)
5
÷ (3
2
)
5
(iii) 9
8
× (x
2
)
5
/ (27)
4
× (x
3
)
2
(iv) 3
2
× 7
8
× 13
6
/ 21
2
× 91
3
Solution:
(i) Given (25)
3
÷ 5
3
= (5
2
)
3
÷ 5
3
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= 5
6
÷
5
3
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 5
6 – 3
= 5
3
(ii) Given (81)
5
÷ (3
2
)
5
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= (81)
5
÷ 3
10
[81 = 3
4
]
= (3
4
)
5
÷ 3
10
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= 3
20
÷ 3
10
= 3
20-10
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 3
10
(iii) Given 9
8
× (x
2
)
5
/ (27)
4
× (x
3
)
2
= (3
2
)
8
× (x
2
)
5
/ (3
3
)
4
× (x
3
)
2
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= 3
16
× x
10
/3
12
× x
6
= 3
16-12
× x
10-6
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 3
4
× x
4
= (3x)
4
(iv) Given (3
2
× 7
8
× 13
6
)/ (21
2
× 91
3
)
= (3
2
× 7
2
7
6
× 13
6
)/(21
2
× 13
3
× 7
3
)[According to the law of exponents we have (a
m
)
n
= a
mn
]
= (21
2
× 7
6
× 13
6
)/(21
2
× 13
3
× 7
3
)
= (7
6
× 13
6
)/(13
3
× 7
3
)
= 91
6
/91
3
[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 91
6-3
= 91
3
6. Simplify:
(i) (3
5
)
11
× (3
15
)
4
– (3
5
)
18
× (3
5
)
5
(ii) (16 × 2
n+1
– 4 × 2
n
)/(16 × 2
n+2
– 2 × 2
n+2
)
(iii) (10 × 5
n+1
+ 25 × 5
n
)/(3 × 5
n+2
+ 10 × 5
n+1
)
(iv) (16)
7
×(25)
5
× (81)
3
/(15)
7
×(24)
5
× (80)
3
Solution:
(i) Given (3
5
)
11
× (3
15
)
4
– (3
5
)
18
× (3
5
)
5
= (3)
55
× (3)
60
– (3)
90
× (3)
25
[According to the law of exponents we have (a
m
)
n
= a
mn
]
= 3
55+60
– 3
90+25
= 3
115
- 3
115
= 0
(ii) Given (16 × 2
n+1
– 4 × 2
n
)/(16 × 2
n+2
– 2 × 2
n+2
)
= (2
4
× 2
(n+1)
-2
2
× 2
n
)/(2
4
× 2
(n+2)
-2
2+1
× 2
2
) [According to the law of exponents we have
(a
m
)
n
= a
mn
]
= 2
2
× 2
(n+3-2n)
/)2
2
× 2
(n+4-2n+1)
= 2
n
× 2
3
– 2
n
/ 2
n
× 2
4
– 2
n
× 2
= 2
n
(2
3
– 1)/ 2
n
(2
4
– 1) [According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 8 -1 /16 -2
= 7/14
= (1/2)
(iii) Given (10 × 5
n+1
+ 25 × 5
n
)/(3 × 5
n+2
+ 10 × 5
n+1
)
= (10 × 5
n+1
+ 5
2
× 5
n
)/(3 × 5
n+2
+ (2 × 5) × 5
n+1
)
= (10 × 5
n+1
+ 5
× 5
n+1
)/(3 × 5
n+2
+ (2 × 5) × 5
n+1
) [According to the law of exponents we
have (a
m
)
n
= a
mn
]
= 5
n+1
(10+5)/ 5
n+1
(10+15)[According to the law of exponents we have a
m
÷ a
n
= a
m-n
]
= 15/25
= (3/5)
(iv) Given (16)
7
×(25)
5
× (81)
3
/(15)
7
×(24)
5
× (80)
3
= (16)
7
×(5
2
)
5
× (3
4
)
3
/(3 × 5 )
7
×(3 × 8)
5
× (16 × 5)
3
= (16)
7
×(5
2
)
5
× (3
4
)
3
/3
7
× 5
7
× 3
5
× 8
5
× 16
3
× 5
3
= (16)
7
/ 8
5
× 16
3
= (16)
4
/8
5
= (2 × 8)
4
/8
5
= 2
4
/8
= (16/8)
= 2
7. Find the values of n in each of the following:
(i) 5
2n
× 5
3
= 5
11
(ii) 9 x 3
n
= 3
7
(iii) 8 x 2
n+2
= 32
(iv) 7
2n+1
÷ 49 = 7
3
(v) (3/2)
4
× (3/2)
5
= (3/2)
2n+1
(vi) (2/3)
10
× {(3/2)
2
}
5
= (2/3)
2n – 2
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FAQs on Exponents (Exercise 6.2) RD Sharma Solutions - Mathematics (Maths) Class 7 (Old NCERT)
| 1. What are exponents? |  |
Ans. Exponents, also known as powers, are mathematical notation used to represent repeated multiplication of a number by itself. They are written as a small superscript number placed to the upper right of a base number.
| 2. How do you read exponents? | |
Ans. Exponents are read in different ways depending on the context. For example, 2^3 can be read as "2 raised to the power of 3" or "2 to the third power". Similarly, 10^2 can be read as "10 squared" or "10 raised to the power of 2".
| 3. What is the meaning of a negative exponent? | |
Ans. A negative exponent indicates the reciprocal of a number or a fraction. For example, 3^-2 is equal to 1/3^2 or 1/9. It means that the number or fraction with a negative exponent is inverted.
| 4. How do you simplify expressions with exponents? | |
Ans. To simplify expressions with exponents, you can use the rules of exponents. If you have the same base raised to different exponents, you can multiply the exponents. For example, (2^3) * (2^2) can be simplified as 2^(3+2) = 2^5.
| 5. What is the difference between a base and an exponent? | |
Ans. The base of an exponent is the number that is multiplied by itself repeatedly. The exponent is the small superscript number that indicates how many times the base is multiplied by itself. For example, in 2^3, 2 is the base and 3 is the exponent.
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