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

7th Grade > Mathematics

EXPONENTS AND POWERS MCQs

Total Questions : 102 | Page 8 of 11 pages

If $x = y^{a}, y = z^{b}, z = x^{c}$ then find abc. If a = 12, b = $\frac{1}{288}$ , then find the value of c. [4 MARKS]

Discuss Question

Answer: Option A. ->
:
Steps: 1 Marks
Application: 1 Mark
Value of abc: 1 Mark
Value of c: 1 Mark
It is given that

x = y^{a}, y = z^{b}, z = x^{c}

x = y^{a}

= (z^{b})^{a}

(∵ y = z^{b})

= z^{a b}

= (x^{c})^{a b}

(∵ x^{c} = z)

x = x^{a b c}

∵ ((a^{m})^{n} = a^{m n})

\Rightarrow x^{1} = x^{a b c}

As bases are equal, exponents are equal.

\Rightarrow a b c = 1

Now, it is also given that,
a = 12, b =

\frac{1}{288}

\times

\times

c = 1
c =

\frac{1}{a \times b}

On substituting the values we get:
c =

\frac{1}{12 \times \frac{1}{288}}

c =

\frac{288}{12}

c =

24

Hence, the value of c is

24

Question 72.

In a race Karan covers 1m in $1^{s t}$ second, in the next second he covers twice the distance he covered in $1^{s t}$ second, in $3^{r d}$ second he covers twice the distance which he covered in $2^{n d}$ second and he continues to run like this till six seconds. After that he slows down, as in $7^{t h}$ second he covers only half the distance which he covered in $6^{t h}$ second and in $8^{t h}$ second he covers half the distance which he covered in $7^{t h}$ second and he continuous in that manner. If Karan completes the race in 9 seconds, find the length of given track. [4 MARKS]

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Answer: Option A. ->
:

Application: 1 Mark
Steps: 2 Marks
Answer: 1 Mark
Here are the distances covered by Karan in each second
$1^{s t}$ second = 1m
$2^{n d}$ second = $1 \times 2$ =2m
Hence, the length of the track is 2 m.
$3^{r d}$ second = $2 \times 2$ = 4m
$4^{t h}$ second = $4 \times 2$ = 8m
$5^{t h}$ second = $8 \times 2$ =16m
$6^{t h}$ second = $16 \times 2$ =32m
$7^{t h}$ second = $32 \div 2$ = 16m
$8^{t h}$ second = $16 \div 2$ = 8m
$9^{t h}$ second = $8 \div 2$ = 4m
Length of the given track = (1+2+4+8+16+32+16+8+4) m
$\Rightarrow$ Length of the given track = 91 m
Hence, the length of the track is 91 m.

Question 73.

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

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Answer: Option A. ->
:

$472 = (2) \times (2) \times (2) \times (59)$
$\Rightarrow 472 = 2^{3} \times 59$

Question 74.

A man has many chocolates and he takes $3^{7}$ chocolates and distributes it to $3^{4}$ students. How many chocolates do each student get if each student should get an equal number of chocolates? [2 MARKS]

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Answer: Option A. ->
:

Concept: 1 Mark
Answer: 1 Mark
Given that,
A man has many chocolates and he takes $3^{7}$ chocolates.
He has to distribute them equally between $3^{4}$ students.
The number of chocolates a student gets is:
$= \frac{3^{7}}{3^{4}}$
$= 3^{7 - 4}$ [ $∵ \frac{a^{m}}{a^{n}} = a^{m - n}$ ]
$= 3^{3}$
$= 27$
So, each student will get 27 chocolates.

Question 75.

What is the difference between $4^{3}$ and $3^{2}$ ? [2 MARKS]

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Answer: Option A. ->
:

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 $3^{2}$ is $55$ .

Question 76.

Simplify: [2 MARKS]

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

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Answer: Option A. ->
:
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 77.

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]

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Answer: Option A. ->
:

Steps: 1 Mark
Application: 1 Mark
Answer: 1 Mark
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 78.

The distance between two cities is 3245678 km. Convert it into meters and express in standard form. [2 MARKS]

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Answer: Option A. ->
:

Conversion: 1 Mark
Standard form: 1 Mark
Given distance that the distance between two cities 3245678 km.
Now, 1 km = 1000 m
3245678 km = (3245678 $\times$ 1000) m = 3245678000 m.
Standard form is,
$3245678000 m = 3.245678 \times 10^{9}$

Question 79.

Simplify : [3 MARKS]
$\frac{3^{8} \times 4^{5} \times 3^{2} \times 5^{4} \times 7^{5}}{3^{9} \times 4^{4} \times 7^{5}}$

Discuss Question

Answer: Option A. ->
:

Concept: 1 Mark
Steps: 1 Mark
Answer: 1 Mark
Given,
$\frac{3^{8} \times 4^{5} \times 3^{2} \times 5^{4} \times 7^{5}}{3^{9} \times 4^{4} \times 7^{5}}$
$= \frac{3^{8 + 2} \times 4^{5} \times 5^{4} \times 7^{5}}{3^{9} \times 4^{4} \times 7^{5}}$
[ $∵ a^{m} \times a^{n} = a^{m + n}$ ]
$= 3^{10 - 9} \times 4^{5 - 4} \times 5^{4} \times 7^{5 - 5}$
[ $∵ \frac{a^{m}}{a^{n}} = a^{m - n}$ ]
$= 3^{1} \times 4^{1} \times 5^{4} \times 7^{0}$
$= 3 \times 4 \times 5^{4}$ [ $∵ a^{0} = 1$ ]
$= 7500$