Chapter 8: Exponents and Powers

Ans: Use ${a^m} \times {a^n} = {\left( a \right)^{m + n}}$, ${a^m} \div {a^n} = {\left( a \right)^{m - n}}$ and ${a^{ - m}} = \dfrac{1}{{{a^m}}}$.

$\dfrac{{16 \times {{10}^2} \times 64}}{{{2^4} \times {4^2}}} = {\left( 4 \right)^2} \times {10^2} \times {2^{ - 4}} \times {4^3} \times {4^{ - 2}}$

= ${\left( {{2^2}} \right)^2} \times {10^2} \times {\left( {{2^2}} \right)^{ - 2}} \times {4^3} \times {4^{ - 2}}$

= ${4^2} \times {10^2} \times {4^{ - 2}} \times {4^3} \times {4^{ - 2}}$

= ${4^{2 - 2 + 3 - 2}} \times {10^2}$

= ${4^1} \times {10^2} = 400$

Ans: Use ${a^m} \times {a^n} = {\left( a \right)^{m + n}}$, ${a^m} \div {a^n} = {\left( a \right)^{m - n}}$ and ${a^{ - m}} = \dfrac{1}{{{a^m}}}$.

$\Rightarrow {5^{m + 3 + \left( { - 2} \right) - \left( { - 5} \right)}} = {5^{12}}$

$\Rightarrow {5^{m + 3 - 2 + 5}} = {5^{12}}$

$\Rightarrow {5^{m + 6}} = {5^{12}}$

On comparing both sides,

⇒m + 6 = 12

⇒n = 6

Ans: Weight of new born bear = 4 kg
Weight increases by the power of 2 in 5 yr.
Weight of bear in 5 yr = (4)2 = 16 kg

Ans: The bacteria cell in every 30 min= 2

So, cell of a bacterium in 1h = 22 $[$As (2+2) in 1h$]$

cell of a bacterium in 12h = 22 x 22 x 22 x 22 x22 x 22 x 22 x

22 x 22 x 22 x 22 x 22 = 224 

Ans: Similarly, bacteria in 24 h = 224 x 224 = 224+24 = 248

Ans: Distance between planet A and Earth = 9.35 x 106 km Distance between planet B and Earth = 6.27 x 107 km
For finding difference between above two distances, we have to change both in same exponent of 10, i.e. 9.35 x 106 = 0.935 x 107, clearly 6.27 x 107 is greater.
So, planet A is nearer to Earth.

Ans: The cell of a bacteria double itself every hour = 1 + 1

= 2 = 21

Since, the process started with 1 cell.

As, am × an = am+n

Therefore, total no. of cell in 8h = 21 x 21 x 21 x 21 x 21 x 21 x

21 x 21 = 28

Ans: On the basis of given information in the question, we can arrange the following table which shows the insect’s location for the first 10 hops.

Ans: If we see the distance covered in each hops

Distance covered in 1st hops = $1-\dfrac{1}{2}$

Distance covered in 2nd hops = $1-\dfrac{1}{4}$

Distance covered in 3rd hops = $1-\dfrac{1}{8}$

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Distance covered in n hops = $1-{\left( {\dfrac{1}{2}} \right)^n}$

Ans: No, because for reaching 1, ${\left( {\dfrac{1}{2}} \right)^n}$ has to be zero for finite n

which is not possible.

Ans: On the basis of given pattern in 1x and 2x , we can make more patterns for 3x 4x , 5x ,6x , 7x , 8x , 9x , 10x .
Thus, we have the following table which shows all details about the patterns.

(b) Predict the ones digit in the following:

1. 412                  

Ans: Ones digit in 412 is 6.

2. 920   

Ans: Ones digit in 920 is 1.

3. 317

Ans: Ones digit in 317 is 3.

 4. 5100 

Ans: Ones digit in 5100 is 5.

 4. 10500 

Ans: Ones digit in 10500 is 0.

(c) Predict the ones digit in the following:

1. 3110                

Ans: Ones digit in 3110 is 1.

2. 1210 

Ans:  Ones digit in 1210 is 4.

3. 1721    

Ans:  Ones digit in 1721 is 7.

4. 2910

Ans:  One digit in 2910 is 1.

Ans: Using law of exponents am × an = am+n

Sun= 199 x 1028 = 1.99 x 1028 x 102 = 1.99 x 1030

Mercury= 33 x 1022 = 3.3 x 1022 x 10 = 3.3 x 1023

Venus= 487 x 1022 = 4.87 x 1022 x 102 = 4.87 x 1024

Earth= 597 x 1022 = 5.97 x 1022 x 102 = 5.97 x 1024

Mars= 642 x 1027 = 6.42 x 1027 x 102 = 6.42 x 1029

Jupiter= 19 x 1026 = 1.9 x 1026 x 10= 1.9 x 1027

Saturn= 568 x 1024 = 5.68 x 1024 x 102 = 5.68 x 1026

Uranus= 868 x 1023 = 8.68 x 1023 x 102 = 8.68 x 1025

Neptune= 102 x 1024 = 1.02 x 1024 x 102 = 1.02 x 1026

Pluto= 127 x 1024 = 1.27 x 1020 x 102 = 1.27 x 1022

Moon= 795 x 1020 = 7.95 x 1020 x 102 = 7.95 x 1022

Ans: Order of mass of all planets and Moon from least to greatest.

Pluto < Moon < Mercury < Venus < Earth < Uranus < Neptune < Saturn < Jupiter < Mars

Ans: Venus has the mass similar to Earth.

Ans: Scientific notation of distance from Sun to

Earth= 149600000 = 1496 x 105 = 1.496 x 108

Jupiter= 778300000 = 7783 x 105 = 7.783 x 108

Mars= 227900000 = 2279 x 105 = 2.279 x 108

Mercury= 57900000 = 579 x 105 = 5.97 x 107

Neptune= 4497000000 = 4497 x 105 = 4.497 x 108

Pluto= 5900000000 = 59 x 108 = 5.9 x 109

Saturn= 1427000000= 1427 x 106 = 1.427 x 109

Venus= 108200000 = 1082 x 105 = 1.082 x 108

Ans: Order of planet from closest to the Sun to farthest from the

sun is given by

Mercury < Venus < Earth < Mars < Jupiter < Saturn < Uranus

< Neptune < Pluto.