Exponents And Powers(7th Grade > Mathematics ) Questions and answers for exam Preparation

7th Grade > Mathematics

EXPONENTS AND POWERS MCQs

Total Questions : 102 | Page 4 of 11 pages

Question 31. Simplify:

\frac{8^{4} \times 14^{3} \times 5 \times 3^{4}}{4^{5} \times 7^{3} \times 6^{2}}

[3 MARKS]

Discuss Question

:
Application: 1 Mark
Steps: 1 Mark
Answer: 1 Mark
Consider,

\frac{8^{4} \times 14^{3} \times 5 \times 3^{4}}{4^{5} \times 7^{3} \times 6^{2}}

8^{4} = (2^{3})^{4} = 2^{12}

14^{3} = (2 \times 7)^{3} = 2^{3} \times 7^{3}

4^{5} = (2^{2})^{5} = 2^{10}

6^{2} = (2 \times 3)^{2} = 2^{2} \times 3^{2}

On substituting the values we get:

\frac{2^{12} \times 2^{3} \times 7^{3} \times 5 \times 3^{4}}{2^{10} \times 7^{3} \times 2^{2} \times 3^{2}}

= 2^{12 + 3 - 10 - 2} \times 7^{3 - 3} \times 5 \times 3^{4 - 2}

[

∵ \frac{a^{m}}{a^{n}} = a^{m - n}

]

= 2^{3} \times 5 \times 3^{2}

[

∵ a^{0} = 1

]

= 8 \times 5 \times 9

= 360

Question 32. Simplify: [2 MARKS]

2^{5} \times 3 \times 4^{2} \times 5

Discuss Question

:
Steps: 1 Mark
Answer: 1 Mark
Given equation is:

2^{5} \times 3 \times 4^{2} \times 5

= 2^{5} \times 3 \times (2^{2})^{2} \times 5

= 2^{5} \times 3 \times 2^{4} \times 5

∵ (a^{m})^{n} = a^{m n}

= 2^{5 + 4} \times 3 \times 5

∵ a^{m} \times a^{n} = a^{m + n}

= 2^{9} \times 3 \times 5

= 2^{9} \times 3 \times 5

= 7680

Question 33. In a stck there are 5 books each of thickness 20mm and 5 paper sheets of thickness 0.016mm. What is the total thickness of the stack? (1mm = 0.001m) [3 MARKS]

Discuss Question

:
Steps: 1 Mark
Application: 1 Mark
Answer: 1Mark
Thickness of each book

= 20 m m

Hence, thickness of 5 books

= 5 \times 20 = 100 m m

Thickness of each paper sheet

= 0.016 m m

Hence, thickness of 5 paper sheets

= 5 \times 0.016 = 0.080 m m

Total thickness of the stack
= Thickness of 5 books + Thickness of 5 paper sheets

= (100 + 0.080) m m

= 100.08 m m

= 1.0008 \times 10^{2} m m

Question 34. Simplify:

(i) 0.005 \times 10^{2} - 5 \times 10^{- 1}

(i i) \frac{(2^{5})^{2} \times 7^{3}}{8^{3} \times 7}

[4 MARKS]

Discuss Question

:
Each question: 2 Marks
i) It's easy to subtract powerswhen we convert theminto normal form.

0.005 \times 10^{2} = 0.5

(move the decimal point 2 places to the right)

5 \times 10^{- 1} = 0.5

(move the decimal point 1placeto the left)
Now,

0.005 \times 10^{2} - 5 \times 10^{- 1}

= 0.5 - 0.5

= 0

ii)

\frac{(2^{5})^{2} \times 7^{3}}{8^{3} \times 7}

\frac{(2^{5})^{2} \times 7^{3}}{8^{3} \times 7}

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

\frac{2^{10} \times 7^{3}}{2^{9} \times 7}

2^{10 - 9} \times 7^{3 - 1} = 2 \times 7^{2}

2 \times 49 = 98

Question 35. Express 472 as a power of its prime factors. [1 MARK]

Discuss Question

472 = (2) \times (2) \times (2) \times (59)

\Rightarrow 472 = 2^{3} \times 59

Question 36. The value of a man's property is greater than the multiplication of cube and square of properties of A and B respectively.
What is the minimum value of his property, if the value of properties A and B are Rs 20 and Rs 400 respectively?
The actual value of his property was Rs 2,00,00,00,000 but due to recession, the value of the property decreased by

30 %

.
Find the present value of the property. [4 MARKS]

Discuss Question

:
Steps: 2 Marks
Minimum Value: 1 Mark
Present Value: 1 Mark
Given that,
A's property =Rs 20

\Rightarrow

cube ofA's property

= 20^{3}

B's property =Rs 400

\Rightarrow

Squareof B's property

= 400^{2}

The minimum value of the man'sproperty should be

= R s 20^{3} \times 400^{2}

= R s 128 \times 10^{7}

= R s 1.28 \times 10^{9}

Given that
The actual value of the property isRs2,00,00,00,000.
The value of the property depreciated by

30 %

.
The present value of the property

= 2000000000 - 2000000000 \times \frac{30}{100}

= 2000000000 - 600000000

= 1400000000

Hence, the present value of the property is

R s 1400000000

Question 37. Write the following numbers in the expanded form:
(i) 279404
(ii) 3006194
(iii) 2806196
[3 Marks]

Discuss Question

:
Each option: 1 Mark
i) 279404 = 2,00,000 + 70,000 + 9,000 + 400 + 00 +4
=

2 \times 100000 + 7 \times 10000 + 9 \times 1000 + 4 \times 100 + 0 \times 10 + 4 \times 1

2 \times 10^{5} + 7 \times 10^{4} + 9 \times 10^{3} + 4 \times 10^{2} + 0 \times 10^{1} + 4 \times 10^{0}

ii) 3006194 = 30,00,000 + 0 + 0 + 6,000 + 100 + 90 + 4
=

3 \times 1000000 + 0 \times 100000 + 0 \times 10000 + 6 \times 1000 + 1 \times 100 + 9 \times 10 + 4 \times 1

3 \times 10^{6} + 0 \times 10^{5} + 0 \times 10^{4} + 6 \times 10^{3} + 1 \times 10^{2} + 9 \times 10^{1} + 4 \times 10^{0}

iii) 2806196 = 20,00,000 + 8,00,000 + 0 + 6,000 + 100 + 90 + 6
=

2 \times 1000000 + 8 \times 100000 + 0 \times 10000 + 6 \times 1000 + 1 \times 100 + 9 \times 10 + 6 \times 1

2 \times 10^{6} + 8 \times 10^{5} + 0 \times 10^{4} + 6 \times 10^{3} + 1 \times 10^{2} + 9 \times 10^{1} + 6 \times 10^{0}

Question 38. Simplify and express in exponential form:

[(5^{2})^{3} \times 5^{4}] \div 5^{7}

[3 MARKS]

Discuss Question

:
Steps: 2 Marks
Answer: 1 Mark

= [(5^{2})^{3} \times 5^{4}] \div 5^{7}

= [5^{6} \times 5^{4}] \div 5^{7}

[∵ (a^{m})^{n} = a^{m \times n}]

= [5^{6 + 4}] \div 5^{7}

[∵ a^{m} \times a^{n} = a^{m + n}]

= 5^{10} \div 5^{7}

= 5^{10 - 7}

[∵ a^{m} \div a^{n} = a^{m - n}]

= 5^{3}

Question 39. What is the difference between

4^{3}

and

3^{2}

? [2 MARKS]

Discuss Question

:
Steps: 1 Mark
Answer: 1 Mark

4^{3} = (4) \times (4) \times (4)

\Rightarrow 4^{3} = 64

3^{2} = (3) \times (3)

\Rightarrow 3^{2} = 9

The difference between given numbers is:

= 64 - 9 = 55

The difference between

4^{3}

and

3^{2}

55

Question 40. Find the value of 'a' satisfying the equation.
a)

{(4^{2})}^{a} = {(7^{a})}^{2}

.
b)

(2^{a})^{5} = 2^{4} \times 4^{3}

[4 MARKS]

Discuss Question

:
Each option: 2 Marks
a) Given that,

{(4^{2})}^{a} = {(7^{a})}^{2}

4^{2 a} = 7^{2 a}

[{a^{m}}^{n} = a^{m n}]

Given,

4^{2 a} = 7^{2 a}

Since

4 \neq 7

the only case where the above equation is true is when both the exponents are zero.

\Rightarrow a = 0

\Rightarrow 4^{2 a} = 7^{2 a} = 1

[∵ 4^{0} = 1, 7^{0} = 1]

b) The given equation is

(2^{a})^{5} = 2^{4} \times 4^{3}

\Rightarrow 2^{a \times 5} = 2^{4} \times (2^{2})^{3}

∵ (a^{m})^{n} = a^{m \times n}

\Rightarrow 2^{a \times 5} = 2^{4} \times (2^{2 \times 3})

\Rightarrow 2^{a \times 5} = 2^{4} \times (2^{6})

\Rightarrow 2^{a \times 5} = 2^{4 + 6} = 2^{10}

∵ a^{m} \times a^{n} = a^{m + n}

Since their bases are same andthey are equal, threforetheir powers must be same,
So,

5 \times a = 10

\Rightarrow a = \frac{10}{5}

\Rightarrow a = 2

So, the value of a = 2.

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EXPONENTS AND POWERS MCQs

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