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Given the speed of light = 30,00,00,000 m/sec 

= 3 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 

∴ Speed of light = 3 × 10⁸ m/sec.

The above statement is true.

LHS = (ab)^m = (a × b)^m = a^m × b^m ……….(by law of exponents)

(1)¹²¹ = -1 × -1 × -1 ……..(121 times) 

Here 121 is an odd number

So, (1)¹²¹= – 1

Given population of India = 121,00,00,000 

= 121 × 10 × 10 × 10 × 10 × 10 × 10 × 10

= 11 × 11 × 10⁷

∴ Population of India = 11² × 10⁷ as per 2011 census.

True

e.g. 2360000 

= 236 x 10 x 10 x 10 x 10

= 236 x 10⁴

= 2.36 x 10⁴ x 10²

=2.36 x 10⁶

True.

We know that, x⁰ = 1

so, x⁰ × x⁰ = 1 1 = 1

x⁰ ÷ x⁰ = 1 ÷ 1 = 1

Therefore, x⁰ × x⁰ = x⁰ ÷ x⁰

1 = 1

False

Any number can be expressed as a decimal number between 1.0 and 10.0 (including 1.0) multiplied by a power of 10. Such form of a number is called its standard form or scientific notation.

Given that, 5 hectares = 5 × 10000 m² (∵1 hectare = 10000 m²)

= 5 × 10000 × 100 × 100 cm²

Standard form = 5 × 10000 × 100 × 100

= 5 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10

Using the law of exponent, a^m x aⁿ = a^m+n

= 5 × 10⁸ cm²

∴ The standard form is 5 × 10⁸ cm²

False.

4² = 4 × 4 = 16

2⁴ = 2 × 2 × 2 × 2 = 16

Therefore, 4² = 2⁴

In order to find the number, which should divide (- 4)⁵ to get the quotient (- 4)³, we will divide (- 4)⁵ by (- 4)³.

Hence, required number = = (- 4)^5-3 = (-4)²

Let as assume the number is x.

Let x be multiply with (–29)⁰ and the product becomes ( + 29)⁰

So, x x (29)⁰ = (+29)⁰

Using the law of exponent, a⁰ = 1

⇒ x × 1 = 1

∴ x = 1

∴The number is 1

Given that, 5^3x–1 ÷ 25 = 125

∵ 25 = 5 × 5 = 5² and 25 = 5 × 5 × 5 = 5³

∴ 5^3x–1 ÷ 5² = 5³

Using the law of exponent, a^m/aⁿ = a^m-n

⇒ 5^3x-1-2 = 5³

⇒ 5^3x-3 = 5³

Comparing both sides,

3x-3 = 3

⇒3x = 6

⇒x = 2

∴ The value of x is 2

3⁷ × 3⁸

We know a^m × aⁿ = a ^m+n 

3⁷ × 3⁸ = 3⁷ + 8 = 3¹⁵ 

∴ 3⁷ × 3⁸ =3¹⁵

7⁵ × 7^3x = 7²⁰

W know a^m x aⁿ = a^m+n

7^5+3x = 7²⁰

If the bases are equal, then the powers should also be equal. 

⇒ 5 + 3x = 20 

⇒ 5 + 3x – 5 = 20 – 5 

⇒ 3x = 15

⇒ 3x/3 = 15/3 = 5

∴ x = 5

(10000)⁰

we know a⁰ = 1 

So, (10000)⁰ = 1

Given, 10^y = 10000 

10^y = 10 × 10 × 10 × 10 

10^y = 10⁴ 

If the bases are equal, then the powers should also be equal. 

⇒ y = 4

Multiply by 5 on both sides, 

5 × y = 5 × 4 

∴ 5y = 20

9² × 9⁰ × 9³

We know a^p × a^q × a^r = p + q + r 

9² × 9⁰ × 9³ = 9²⁺⁰⁺³ = 9⁵

9² × 9⁰ × 9³ = 9⁵

According to the question, in a shrinking machine, a piece of stick is compressed to reduce its length. If 9 cm long sandwich is put into the shrinking machine, then the length of sandwich will be 9 x 1/ 3^-1= 9 x 3 = 27 cm.

According to the question, if we put a 3 cm piece of wire through a (x 2⁴) machine, its length becomes 3 x 2 x 2 x 2 x 2 = 48 cm.

Similarly, 4 cm long strip becomes 4 x 2 x 2 x 2 x 2 = 64 cm.

According to the question, if we put a 5 m stick into a (x 4) stretching machine, then machine produces 20 m stick.

Similarly, if we put 10 cm carrot into a (x 4) stretching machine, then machine produce 10 x 4= 40 cm stick.

Since, 5² = 25, 5³ = 125, 5⁴ = 625

Hence, (x 5²), (x 5³)and (x 5⁴) machine can replace (x 5) hook-up machine.

Factorization of 1/8 = 1/2 x 1/2 x 1/2 = 1/2³

Therefore, a (x 1/2³) machine can do the same work as a (x 1/8 ) machine.

(2⁸)³

We know (a^m)ⁿ = a^mn

(2⁸)³ = 2^8x3 =2²⁴ 

∴ (2⁸)³ = 2²⁴ 

Correct option is (A) 6

\(9^x-9^{x-1}=72\)

\(\Rightarrow\) \(9^{x-1}(9-1)=72\)

\(\Rightarrow\) \(9^{x-1}=\frac{72}8=9\)

\(\Rightarrow\) x - 1 = 1

\(\Rightarrow\) x = 1 - (-1) = 1+1 = 2

\(\therefore\) 3x = 3 \(\times\) 2 = 6

Correct option is  A) 6

Correct option is (A) 3¹⁰

Let we multiply x with \(3^{-4}\) to get 729.

\(\therefore\) \(x\times3^{-4}=729=3^6\)

\(\Rightarrow\) \(x=\frac{3^6}{3^{-4}}=3^6\times(3^{-4})^{-1}\) \(=3^6\times3^4=3^{6+4}=3^{10}\)

Thus, we have to multiply \(3^{10}\) with \(3^{-4}\) to get 729.

Correct option is  A) 3¹⁰ 

(a⁵)⁴

We know (a^m)ⁿ = a^mn

(a⁵)⁴ = a^5x4 =a²⁰ 

∴ (a⁵)⁴ = a²⁰ 

Correct option is (D) 9

\(\frac{1}{81}th\) part (value) of 729 \(=\frac{1}{81}\times729=9\)

Correct option is  D) 9

Length of the paper clip = 0.05 m

In standard form, 0.05 m = 0.5 x 10^-1 = 5.0 x 10^-2 m

Hence, the length of the paper clip in standard form is 5.0 x 10^-2 m

The radius of a proton is 1.3 fermi.

One fermi is equal to 10^-15 m.

So, the radius of the proton is 1.3 x 10^-15 m.

Hence, standard form of radius of the proton is 1.3 x  10^-15 m.

Total seconds in a day = 86400

So, a second is long as 1/86400 = 0.000011574

Scientific notation of 0.000011574= 1.1574 x 10^-5 days

(i) 5,00,00,000 = 5 x 10000000 = 5 x 10⁷

(ii) 70,00,000 = 7 x 1000000 = 7 x 10⁶

Number of molecules in a dot of water of weight 1.8 gm

= 60. 230, 000, 000, 000, 000, 000, 000

= 6.023 x 1000000000000000000000

= 6.023 x 10²²

Given mass of electron = 9.1093826 × 10^-31 kilograms

We know that 1 kilogram = 1000 grams = 10³ grams.

⇒ Mass of electron = 9.1093826 × 10^-31 × 10³

We know that by properties of exponents, a^m × aⁿ = a^{m + n}.

⇒ 9.1093826 × 10^-31 × 10³ 

= 9.1093826 × 10^{-31 + 3}

= 9.1093826 × 10^-28

∴ Mass of electron in grams = 9.1093826 × 10^-28 grams

We have, 2ⁿ⁺² +2ⁿ⁺¹ + 2n = c x 2ⁿ

=> 2ⁿ 2² +2ⁿ 2¹ + 2ⁿ = c x 2ⁿ [∴ a^m+n = a^m x aⁿ]

=> 2ⁿ [2² – 2¹ + 1] = c x 2ⁿ [taking common 2ⁿ in LHS]

=> 2ⁿ[4-2 + 1]=c x 2ⁿ

=> 3  x 2ⁿ  = c x 2ⁿ

3  x 2ⁿ x 2^-n =c  x 3ⁿ x c^-n [multiplying both sides by 2^-n]

=> 3 x 2^n-1 = c x 2^n-n   [∴ a^m+n = a^m x aⁿ]

=> 3 x 2° =c x 2°

=> 3x 1 =c x 1    [∴ a°=1]

∴  3 = c

(i) (2/3)⁵

(ii) 1,

(iii) 2 x 2 x 2 x 3 x 3

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.

a. Cell of a bacteria in 30 mins = 2 (double in every 30 minutes)

So, cell of bacteria in 1 hour = 2² = 4

Cell of bacteria in 12 hours = 2^{2 x 12}   = 2²⁴

b. Cell of bacteria in 24 hours = 2^{2 x 24}  = 2⁴⁸

Given that, 2 years = 2 × 365 days (∵ 1 year = 365 days)

= 2 × 365 × 24 (∵ 1 day = 24 hours)

= 2 × 365 × 24 × 60 min (∵ 1 hr = 60 min)

= 2 × 365 × 24 × 60 × 60 s (∵ 1 min = 60 s)

= 63072000 s

Standard form = 63072 × 10 × 10 × 10

= 63072 × 10³

= 6.3072 × 10⁴ × 10³

Using the law of exponent, a^m x aⁿ = a^m+n

= 6.3072 × 10⁷ s

∴ The standard form is 6.3072 × 10⁷ s

An urban area is regarded as one which is a town or a city.

False

LHS = 5^o

Using law of exponents, a^o = 1

5^o = 1

Given that,5 tons = 5 × 100 kg (∵ 1 ton = 100 kg)

= 5 × 100 × 1000 g (∵ 1 kg = 1000 g)

= 500000 g

Standard form = 5 × 10 × 10 × 10 × 10 × 10

= 5 × 10⁵

∴ The standard form is 5 × 10⁵ g

Given that, The diameter of a helium atom = 0.000000022 cm

Standard form = 0.22 × 10^-7 cm

= 2.2 × 10^-1 × 10^-7

Using the law of exponent, a^m x aⁿ = a^m+n

= 2.2 × 10^-1-7

= 2.2 × 10^-8 cm

∴ The standard form is 2.2 × 10^-8 cm

The Municipalities have a lot of tasks to perform. Besides Councilors or Corporators, these municipalities employ a large number of workers, officers, clerks, and accountants. Each Municipality has a number of departments each headed by an officer who is responsible for that department. Councilors or Corporators keep in touch with the people of the ward and understand their needs and problems and discuss them in municipal meetings.

Given that,56 km = 56 × 1000 m (∵ 1 km = 1000 m)

= 56000 m

Standard form = 56 × 10³

= 5.6 × 10¹ × 10³

Using the law of exponent, a^m x aⁿ = a^m+n

= 5.6 × 10⁴ m

∴ The standard form is 5.6 × 10⁴ m

Given that, Mass of a molecule of hydrogen gas = 0.00000000000000000000334 tons

Standard form = 0.334 × 10^-20 tons

= 3.34 × 10^-1 × 10^-20

Using the law of exponent, a^m x aⁿ = a^m+n

= 3.34 × 10^-1-20

= 3.34 × 10^-21

∴ The standard form is 3.34 × 10^-21 tons

True

(–7)^–4 × (–7)²

= (-7)^{-4 + 2} 

= (-7)^-2

True

For standard form, 0.000037 

= 0.37 x 10^-4

= 3.7 x 10^-5

(a) (3/4÷5/3)⁵

(By law of exponent: (a)^m÷(b)^m = (a÷b)^m

Correct option is (A) 2

\(9^x+9^{x-1}\) = 90

\(\Rightarrow\) \(9^{x-1}(9+1)=90\)

\(\Rightarrow\) \(9^{x-1}\times10=90\)

\(\Rightarrow\) \(9^{x-1}=\frac{90}{10}=9\)

\(\Rightarrow\) x - 1 = 1

\(\Rightarrow\) x = 1 - (-1) = 1+1 = 2

Correct option is  A) 2

(a) p⁵

(By law of exponent: (a)^m÷(a)ⁿ = (a)^m-n)