Exponents And Powers

According to the law of indices:

(a^m)ⁿ = a^{(m × n)}

Applying the same law here,

In the above question,

a = -3

m = 2

n = 3

∴ m × n = 2 × 3

= 6

[(–3)²]³ = (–3)⁶

So correct answer is B.

Again we have another law of indices which is as follows:

a^m ÷ aⁿ = a ^(m-n)

Here in the question,

a = x

m = 8

n = 2

∴ m – n = 8 – 2

= 6

x⁸ ÷ x² = x⁶

So the correct answer is B.

Let us break the question,

Square of x = x²

Cube of x = x³

According to Law of indices,

a^m × aⁿ = a ^{(m + n)}

Here in the question,

a = x

m = 2

n = 3

∴ m + n = 3 + 2

= 5

x² × x³ = x⁵

So the correct answer is C.

According to Law of Indices,

Here in the question,

a = x

b = y

m = 5

So by the Law of Indices,

x⁵ ÷ y⁵ = (x ÷ y)⁵

So the correct answer is C.

According to Law of indices,

a^m × aⁿ = a ^{(m + n)}

So the correct answer is C.

We have the law of Indices as follows:

a⁰ = 1

Where a = any integer

(1⁰ + 2⁰ + 3⁰) = 1 + 1 + 1

= 3

So the correct answer is C.

The above question can be written as follows:

Again we have another law of indices which is as follows:

a^m ÷ aⁿ = a ^(m-n)

Here in the question,

For first part,

a = 10

m = 22

n = 20

∴ m – n = 22 – 20

= 2

For second part,

a = 10

m = 20

n = 20

∴ m – n = 20 – 20

= 0

We have the law of Indices as follows:

a⁰ = 1

Where a = any integer

So the correct answer is C.

For the standard form notation, only one number is kept before the decimal point and rest all numbers are written after decimal point.

The power of 10 denotes the number of terms shifted after the decimal point.

So for 12345,

Here only one number will be kept before decimal point that is 1.

12345 = 1.2345 × 10⁴

So the correct answer is D.

2¹⁹⁹⁸ – 2¹⁹⁹⁷ – 2¹⁹⁹⁶ + 2¹⁹⁹⁵ = K.2¹⁹⁹⁵

The above question can be written as follows:

For first part,

a = 2

m = 1998

n = 1995

∴ m – n = 1998 - 1995

= 3

For second part,

a = 2

m = 1997

n = 1995

∴ m – n = 1997 – 1995

= 2

For third part,

a = 2

m = 1996

n = 1995

∴ m – n = 1996 - 1995

= 1

For second part,

a = 2

m = 1995

n = 1995

∴ m – n = 1995 – 1995

= 0

k = 8 - 4 – 2 + 1

k = 3

So the correct answer is C.

We have the law of Indices as follows:

a⁰ = 1

Where a = any integer

Let us try option A.

2⁰ + 3⁰ + 4⁰ = 1 + 1 + 1

= 3

2⁰ × 3⁰ × 4⁰ = 1 × 1 × 1

= 1

(3⁰ – 2⁰) × 4⁰ = (1-1) × 1

= 0 × 1

= 0

(3⁰ – 2⁰) × (3⁰ + 2⁰) = (1-1) × (1 + 1)

= 0 × 1

= 0

So the correct answer is B.

For the standard form notation, only one number is kept before the decimal point and rest all numbers are written after decimal point.

The power of 10 denotes the number of terms shifted after the decimal point.

So for 72105.4,

Here only one number will be kept before decimal point that is 7.

Number of terms to be shifted after decimal point = 4

72105.4 = 7.21054 × 10⁴

So n = 4

So the correct answer is C.

So the correct answer is D.

So the correct answer is C.

Any negative number when raised to a power which is an even number, the result obtained is positive power of that number.

Any negative number when raised to a power which is an add number, the result obtained is negative power of that number.

So since in the question, the power is 4, the result of (-5)⁴ will be positive 4^th power of 5.

So the answer will be positive.

By looking at the option, Option C does not match to the desired result as the negative sign will persist in the result making it negative.

So option C does not match with the result.

So correct answer is C.

Let us solve each option,

For option A,

For option B,

Again we have another law of indices which is as follows:

a^m ÷ aⁿ = a ^(m-n)

According to Law of indices,

a^m × aⁿ = a ^{(m + n)}

For option C,

For Option D,

So the correct answer is D.

According to law of indices:

a^m × b^m = (ab)^m

Here in the question,

a = 2/3

b = 5/7

m = 3

So the correct answer is C.

For the standard form notation, only one number is kept before the decimal point and rest all numbers are written after decimal point.

The power of 10 denotes the number of terms shifted after the decimal point.

So for 829030000,

Here only one number will be kept before decimal point that is 8.

Number of terms to be shifted after decimal point = 8

829030000 = 8.2903 × 10⁴

So the correct answer is D.

The options C and D results in very low values.

The option B = 0.001

Option A is = 0.0001

So after seeing all the options, option B has highest value.

So the correct answer is B.

Let us first understand the number of zeroes in a crore,

One lakh = 1, 00,000

Number of zeroes = 5

Ten Lakhs = 10, 00, 000

Number of zeroes = 6

One crore = 1, 00, 00, 000

Number of zeroes = 7

Ten crore = 10, 00, 00, 000

Number of zeroes = 8

Therefore 72 crore = 72, 00, 00, 000

Number of zeroes = 7

Number of terms to be shifted after decimal point = 8

72, 00, 00, 000 = 7.2 × 10⁸

So the correct answer is C.

Again we have another law of indices which is as follows:

a^m ÷ aⁿ = a ^(m-n)

Here in the question,

Here a = a/b

Therefore, by the law of indices:

So the correct answer is C.

Let us evaluate each option,

For option A,

3²>2³

9>8

This is true.

For option B,

4³>2⁶

64 = 64

This is also true.

For option C,

3³ = 9

27 = 9

This is false.

For option D,

2⁵>5²

32>25

This is also true.

So the correct answer is C.

2⁶ = 64

We know that square of 8 = 64

Therefore second power of 8 is equal to 2⁶.

So the correct answer is B.

According to Law of indices,

a^m × aⁿ = a ^{(m + n)}

Here in the question,

a = -2

m = 31

n = 13

∴ m + n = 31 + 13

= 44

(-2)³¹ × (-2)¹³ = (-2)⁴⁴

Again we have another law of indices which is as follows:

a^m ÷ aⁿ = a ^(m-n)

Here in the question,

a = -3

m = 8

n = 5

∴ m – n = 8 – 5

= 3

(-3)⁸ ÷ (-3)⁵ = (-3)³

According to Law of indices,

a^m × aⁿ = a ^{(m + n)}

Here in the question,

We will have to multiply (11/15)⁵ with (11/15)⁴ to get (11/15)⁹

Therefore the answer for the fill in the blanks = 11/15

According to Law of indices,

a^m × aⁿ = a ^{(m + n)}

a = -1/4

m = 3

m + n = 11

∴ n = 8

Here in the question,

We will have to multiply (-1/4)⁸ with (-1/4)³ to get (-1/4)¹¹

Therefore the answer for the fill in the blanks = 8

According to the law of indices:

(a^m)ⁿ = a^{(m × n)}

Applying the same law here,

In the above question,

a = 7/11

m = 3

n = 4

∴ m × n = 3 × 4

= 12

[(7/11)³]⁴ = (7/11)¹²

According to the law of indices:

(a^m)ⁿ = a^{(m × n)}

Applying the same law here,

In the above question,

a = 6/13

m = 5

n = 2

∴ m × n = 5 × 2

= 10

[(6/13)⁵]² = (6/13)¹⁰

Again we have another law of indices which is as follows:

a^m ÷ aⁿ = a ^(m-n)

Here in the question,

a = 6/13

m = 10

n = 10

∴ m – n = 10 - 10

= 0

(6/13)¹⁰ ÷ (6/13)¹⁰ = (6/13)⁰

According to the law of indices:

(a^m)ⁿ = a^{(m × n)}

Applying the same law here,

In the above question,

a = -1/4

m = 16

n = 2

∴ m × n = 16 × 2

= 32

[(-1/4)¹⁶]² = (-1/4)³²

According to Law of indices,

a^m ÷ aⁿ = a ^{(m - n)}

Here in the question,

We will have to divide (13/14)⁵ by (13/14)² to get (13/14)³

Therefore the answer for the fill in the blanks = 13/14

According to Law of indices,

a^m × aⁿ = a ^{(m + n)}

m = 6

n = 5

∴ m + n = 5 + 6

= 11

We have the law of Indices as follows:

a⁰ = 1

Where a = any integer

a⁶ × a⁵ × a⁰ = a¹¹ × 1

= a¹¹

Therefore the answer for the fill in the blanks = 11

The power of ten represent the number of zeroes in 1 lakh.

So we know 1 lakh = 1, 00, 000

So total number of zeroes in 1 lakh = 5

1 lakh = 10⁵

Therefore the answer for the fill in the blanks = 5

The power of ten represent the number of zeroes in 1 million.

So we know 1 million = 1, 000, 000

So total number of zeroes in 1 million = 6

1 million = 10⁶

Therefore the answer for the fill in the blanks = 6

We know that 729 is the cube of 9

729 = 9³

We also know that square of 3 = 9

So 729 = (3²)³

According to the law of indices:

(a^m)ⁿ = a^{(m × n)}

Applying the same law here,

In the above question,

a = 3

m = 2

n = 3

∴ m × n = 2 × 3

= 6

So 729 = 3⁶

Therefore the answer for the fill in the blanks = 6

2⁴ = 2 × 2 × 2 × 2

= 4 × 4

= 16

So when 432 / 16 we get = 27

27 = 3 × 3 × 3

= 3³

So 432 = 2⁴ × 3³

Therefore the answer for the fill in the blanks = 3

For the standard form notation, only one number is kept before the decimal point and rest all numbers are written after decimal point.

The power of 10 denotes the number of terms shifted after the decimal point.

So for 53700000,

Here only one number will be kept before decimal point that is 5.

Number of terms to be shifted after decimal point = 7

53700000 = 5.37 × 10⁷

Therefore the answer for the fill in the blanks = 5.37

For the standard form notation, only one number is kept before the decimal point and rest all numbers are written after decimal point.

The power of 10 denotes the number of terms shifted after the decimal point.

So for 88880000000,

Here only one number will be kept before decimal point that is 8.

Number of terms to be shifted after decimal point = 10

88880000000 = 8.888 × 10¹⁰

Therefore the answer for the fill in the blanks = 8.888

For the standard form notation, only one number is kept before the decimal point and rest all numbers are written after decimal point.

The power of 10 denotes the number of terms shifted after the decimal point.

So for 27500000,

Here only one number will be kept before decimal point that is 2.

Number of terms to be shifted after decimal point = 7

27500000 = 2.75 × 10⁷

Therefore the answer for the fill in the blanks = 7

For the standard form notation, only one number is kept before the decimal point and rest all numbers are written after decimal point.

The power of 10 denotes the number of terms shifted after the decimal point.

So for 340900000,

Here only one number will be kept before decimal point that is 3.

Number of terms to be shifted after decimal point = 8

340900000 = 3.409× 10⁸

Therefore the answer for the fill in the blanks = 8

(a) We know that 3² = 3 × 3

= 9

So 9 < 15

Therefore the answer for the fill in the blanks = <

(b) We know that 3² = 3 × 3

= 9

We also know that 2³ = 2 × 2 × 2

= 8

So 8 < 9

Therefore the answer for the fill in the blanks = <

(c) We know that 7⁴ = 7 × 7 × 7 × 7

= 49 × 49

= 2401

We also know that 5⁴ = 5 × 5 × 5 × 5

= 25 × 25

= 625

So 2401 > 625

Therefore the answer for the fill in the blanks = >

(d) The power of 10 represent the number of zeroes accompanying 10.

So 10⁵ = 1, 00,000

10,000 < 10⁵

Therefore the answer for the fill in the blanks = <

(e) We know that 6³ = 6 × 6 × 6

= 6 × 36

= 216

We also know that 4⁴ = 4 × 4 × 4 × 4

= 16 × 16

= 256

So 216 < 256

Therefore the answer for the fill in the blanks = <

The power of 10 represent the number of zeroes accompanying 10.

So 10⁷ = 10, 000,000

But one million = 1, 000,000

So 1 million is not equal to 10⁷

So the statement is false.

We know that,

1 hour = 60 minutes

1 minute = 60 seconds

∴ 1 hour = 60 × 60 seconds

= 60² seconds

Therefore the statement is true.

We have the law of Indices as follows:

a⁰ = 1

Where a = any integer

Therefore 1⁰ = 1

Zero raise to any number is zero.

1⁰ × 0¹ = 1 × 0

= 0

So the statement is false.

Let us first solve LHS,

(–3)⁴ = (-3) × (-3) × (-3) × (-3)

= 9 × 9

= 81

So the statement is false.

We know that 3³ = 3 × 3 × 3

= 3 × 9

= 27

We also know that 4³ = 4 × 4 × 4

= 16 × 4

= 64

So 27 < 64

So the given statement is false.

Any negative number when raised to a power which is an even number, the result obtained is positive power of that number.

Any negative number when raised to a power which is an add number, the result obtained is negative power of that number.

So since in the RHS, the power is 100, the result of (-5)¹⁰⁰ will be positive 100^th power of 5.

Hence RHS is equal to LHS

So the statement is true.

Let us solve LHS,

(10 + 10)¹⁰ = (20)¹⁰

Hence LHS is not equal to RHS

So the statement is false.

Again we have another law of indices which is as follows:

a^m ÷ aⁿ = a ^(m-n)

Here in the question,

a = x

m = 0

n = 0

∴ m – n = 0 – 0

= 0

x⁰ ÷ x⁰ = x⁰

According to Law of indices,

a^m × aⁿ = a ^{(m + n)}

Here in the question,

a = x

m = 0

n = 0

∴ m + n = 0 + 0

= 0

x⁰ × x⁰ = x⁰

So the result obtained in both the question is same.

Hence statement is true.

For the standard form notation, only one number is kept before the decimal point and rest all numbers are written after decimal point.

The power of 10 denotes the number of terms shifted after the decimal point.

A number in standard form is represented as a × 10^k where a lies in the range of 1 to 10.

So the statement is false.

We know that 4² = 4 × 4

= 16

We also know that 2⁴ = 2 × 2 × 2 × 2

= 4 × 4

= 16

So 4² is not greater than 2⁴ rather they are equal.

So the statement is false.

x^m + x^m = 2 x^m

The Law of Indices for powers holds only for multiplication when both the bases are same.

So the statement is false.

According to law of indices:

a^m × b^m = (ab)^m

Here in the question,

a = x

b = y

x^m × y^m = (x × y)^m

So the statement is false.

According to law of indices:

a^m ÷ b^m = (a ÷ b)^m

Here in the question,

a = x

b = y

x^m ÷ y^m = (x ÷ y)^m

So the statement is true.

According to Law of indices,

a^m × aⁿ = a ^{(m + n)}

x^m × xⁿ = x^{m + n}

So the statement is true.

We know that 16³ = 16 × 16 × 16

= 256 × 16

= 4096

We also know that 4⁹ = 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4

= 16 ×16 × 16 × 64

= 2, 62, 144

So 4⁹ is greater than 16³.

So the statement is true.

According to law of indices:

a^m ÷ b^m = (a ÷ b)^m

Here in the question,

a = 2 / 5

b = 5 / 2

So the statement is false.

According to law of indices:

a^m × b^m = (ab)^m

Here in the question,

a = 4 / 3

b = 5 / 7

So In the RHS (4/3) has to be multiplied with (5/7) and not added.

So the statement is false.

According to Law of indices,

a^m ÷ aⁿ = a ^{(m - n)}

Here in the question,

a = 5/8

m = 9

n = 4

m – n = 9 – 4

= 5

So the statement is false.

According to Law of indices,

a^m × aⁿ = a ^{(m + n)}

Here in the question,

a = 7/3

m = 2

n = 5

m – n = 2 + 5

= 7

So the statement is false.

25⁰ = (5²)⁰

125⁰ = (5³)⁰

5⁰ = (5¹)⁰

5⁰ × 25⁰ × 125⁰ = (5¹ × 5² × 5³)⁰

= (5⁶)⁰

(5⁶)⁰ is same as (5⁰)⁶

So the statement is true.

Let us add all the terms:

(8 × 1, 00,000) + (7 × 10,000) + (6 × 1000) + (5× 100) + (4 × 10) + 3

⟹ 8, 00,000 + 70,000 + 6000 + 500 + 40 + 3

⟹ 876543

So the statement is true.

Let us add the terms on RHS,

⟹ (6 × 1, 00,000) + (6× 100)

⟹ 6, 00,000 + 600

⟹ 6, 00, 600

So LHS is not equal to RHS.

So the statement is false.

Let us add the terms on LHS:

⟹ (4 × 1, 00,000) + (3 × 10,000) + (2 × 1000) + 1

⟹ 4, 00,000 + 30,000 + 2000 + 1

⟹ 4, 32, 001

So LHS is not equal to RHS.

So the statement is false.

Let us add the terms on LHS:

⟹ (8 × 10, 00,000) + (2 × 10,000) + (5 × 100) + 9

⟹ 80, 00,000 + 20,000 + 500 + 9

⟹ 80, 20, 509

So LHS is equal to RHS.

So the statement is true.

We have the law of Indices as follows:

a⁰ = 1

Where a = any integer

4⁰ + 5⁰ + 6⁰ = 1 + 1 + 1

= 3

(4 + 5 + 6)⁰ = (15)⁰

= 1

So LHS is not equal to RHS.

So the statement is false.

Let us simplify each term,

2⁵ = 4 × 4 × 2

= 16 × 2

= 32

3³ = 3 × 3 × 3

= 27

2³×2 = 2 × 2 × 2 × 2

= 16

(3³)² = 3^3×2

= 36

= 729

3⁵ = 3 × 3 × 3 × 3 × 3

= 243

4⁰ = 1

2³×3 = 2 × 2 × 2 × 3

= 24

So the ascending order is as follows:

1, 16, 24, 27, 32, 243, 729

Let us simplify each term,

2^{2 + 3} = 2⁵ = 4 × 4 × 2

= 16 × 2

= 32

(2²)³ = (2^{2 × 3})

= 2⁶

= 64

2²×2 = 2 × 2 × 2

= 8

(3⁵) / (3²) = 3³

= 3 × 3 × 3

= 27

3² × 3⁰ = 9 × 1

= 9

2³ × 5² = 8 × 25

= 200

So the ascending order is as follows:

200, 64, 32, 27, 9, 8

According to Law of indices,

a^m ÷ aⁿ = a ^{(m - n)}

Here in the question,

We will have to divide (-4)⁵ by (-4)² to get (-4)³

According to Law of indices,

a^x × a^y = a ^{(x + y)}

Here in the question,

a = 2/9

x = 3

y = 6

∴ x + y = 3 + 6

= 9

2m – 1 = 9

2m = 10

m = 5

Reciprocal of the rational number = 32 / 27

(a) We have the law of Indices as follows:

a⁰ = 1

Where a = any integer

7⁰ = 1

(b) According to Law of indices,

a^x ÷ a^y = a ^{(x - y)}

Here in the question,

a = 7

x = 7

y = 7

∴ x - y = 7 - 7

= 0

7⁷ ÷ 7⁷ = 7⁰ = 1

(c) (–7)^{2 × 7 – 6 – 8}

= (–7)^14-14

= (–7)⁰

= 1

(d) (2⁰ + 3⁰ + 4⁰) (4⁰ – 3⁰ – 2⁰) = (1 + 1 + 1) × (1-1-1)

= 3 × -1

= -3

(e) Let us evaluate the terms separately,

2 × 3 × 4 = 6 × 4

= 24

2⁰ × 3⁰ × 4⁰ = 1 × 1 × 1

= 1

2 × 3 × 4 ÷ 2⁰ × 3⁰ × 4⁰ = 24 / 1

= 24

(f) (8⁰ – 2⁰) × (8⁰ + 2⁰)

= (1 – 1) × (1 + 1)

= 0 × 1

= 0

………………….

According to law of indices:

a^m × b^m = (ab)^m

Now comparing on both sides

2^n-5 = 2^-5

n-5 = -5

n = 0

6^2n-4 = 6^-4

2n-4 = -4

2n = -4 + 4

2n = 0

n = 0

(a) For the standard form notation, only one number is kept before the decimal point and rest all numbers are written after decimal point.

The positive power of 10 denotes the number of terms shifted after the decimal point.

8.01 × 10⁷ = 8, 01, 00, 000

(b) For the standard form notation, only one number is kept before the decimal point and rest all numbers are written after decimal point.

The positive power of 10 denotes the number of terms shifted after the decimal point.

1.75 × 10^–3 = 0.00175

(a) 2⁵ = 2 × 2 × 2 × 2 × 2

= 4 × 4 × 2

= 16 × 2

= 32

(b) (-3)⁵ = (-3) × (-3) × (-3) × (-3) × (-3)

= 9 × 9 × (-3)

= 81 × -3

= -243

(c) - (-4)⁴ = - [(-4) × (-4) × (-4) × (-4)]

= - [16 × 16]

= - 256

(a) Number of terms of 3 = 3

Number of terms of a = 4

3 × 3 × 3 × a × a × a ×a = 3³a⁴

(b) Number of terms of a = 2

Number of terms of b = 3

Number of terms of c = 4

a × a × b × b × b × c × c × c × c = a²b³c⁴

(c) Number of terms of s = 4

Number of terms of t = 3

s × s × t × t × s × s × t = s⁴t³

Number of times 30 should be added = 30⁷ / 30

= 30⁶

To understand the above method,

Take example of 2⁴

2⁴ = 16

But when we add 2 eight times we get 16 [16 / 2].

(a) 1024 = 32 × 32

= 2⁵ × 2⁵

= 2¹⁰

(b) 1029 = 32 × 32 + 5¹

= 2⁵ × 2⁵ + 5¹

= 2¹⁰ + 5¹

(c)

(a) Let us find the values of each term,

2⁶ = 2 × 2 × 2 × 2 × 2 × 2

= 4 × 4 × 4

= 64

6² = 36

So 2⁶ is greater.

(b) 2⁹ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

= 8 × 8 × 8

= 512

9² = 81

So 2⁹ is greater.

(c) 7.9 × 10⁴ = 79, 000

5.28 × 10⁵ = 5, 28, 000

So it is clearly seen that 5.28 × 10⁵ is greater

(a) 9000 = 5 × 5 × 5 × 3 × 3 × 2 × 2 × 2

9000 = 5³ × 3² × 2³

(b) 2025 = 5 × 5 × 3 × 3 ×3 × 3

2025 = 5² × 3⁴

(c) 800 = 2 × 2 × 2 × 10 × 10

800 = 2 × 2 × 2 × 5 × 2 × 5 × 2

800 = 2⁵ × 5²

(a) According to law of indices:

a^m × b^m = (ab)^m

Here in the question,

2³ × 3³ = (3 × 2)³

= 6³

(b) 2³ = 2 × 2 × 2 × 2

= 16

4² = 16

2⁴ × 4² = 16 × 16

= 16²

(c) According to law of indices:

a^m × b^m = (ab)^m

Here in the question,

5² = 25

7² = 49

5² × 7² = (5 × 7)²

= 35²

(d) (– 5)⁵ × (–5) = (– 5)^{5 + 1}

= (– 5)⁶

(e) According to law of indices:

a^m × b^m = (ab)^m

Here in the question,

(– 3)³ × (– 10)³ = (-3 × -10)³

= (30)³

(f) According to law of indices:

a^m × b^m = (ab)^m

Here in the question,

(– 11)² × (– 2)² = (-11 × -2)²

= 22²

For the standard form notation, only one number is kept before the decimal point and rest all numbers are written after decimal point.

The power of 10 denotes the number of terms shifted after the decimal point.

So for 76, 47, 000,

Here only one number will be kept before decimal point that is 7.

Number of terms to be shifted after decimal point = 6

76, 47, 000 = 7.647 × 10⁶

For the standard form notation, only one number is kept before the decimal point and rest all numbers are written after decimal point.

The power of 10 denotes the number of terms shifted after the decimal point.

So for 8, 19, 00, 000,

Here only one number will be kept before decimal point that is 8.

Number of terms to be shifted after decimal point = 7

8, 19, 00, 000 = 8.19 × 10⁷

For the standard form notation, only one number is kept before the decimal point and rest all numbers are written after decimal point.

The power of 10 denotes the number of terms shifted after the decimal point.

So for 5, 83, 00, 00, 00, 000,

Here only one number will be kept before decimal point that is 5.

Number of terms to be shifted after decimal point = 11

5, 83, 00, 00, 00, 000 = 5.83 × 10¹¹

For the standard form notation, only one number is kept before the decimal point and rest all numbers are written after decimal point.

The power of 10 denotes the number of terms shifted after the decimal point.

So for 24 billion = 24,000,000,000

Here only one number will be kept before decimal point that is 2.

Number of terms to be shifted after decimal point = 10

24,000,000,000 = 2.4 × 10¹⁰

Given:

Speed of light = 3 × 10⁸ m/s

Time Taken = 8 minutes

= 8 × 60

= 480 seconds

Speed = Distance / Time

Distance = Speed × Time

Distance = (3 × 10⁸) × (480)

= 1.44 × 10¹¹ m

[As we know that: a^m ÷ aⁿ = a^m-n]

Therefore,

(3⁷ ÷ 3⁵)⁴

⟹ (3^7-5)⁴

⟹ (3²)⁴

⟹ 3⁸

⟹ 49a²b³

⟹ 5^(4-3) × 7^(4-2) × 2^(7-3)

⟹ 5¹ × 7² × 2⁴

⟹ 5 × 49 × 16

= 3920

⟹ 3⁴ × 3²

= 3⁶

= 729

= 1

The ascending order is as follows:

1.554 × 10⁵; 1.9943 × 10⁵; 6.475 × 10⁵; 9.324 × 10⁵; 8.5988 × 10⁶

Gibson, Australia; Thar, India; Great Victoria, Australia, Kalahari, South Africa, Sahara, North Africa

The descending order is as follows:

1.9 × 10²⁷; 5.69 × 10²⁶; 1.02 × 10²⁶; 8.69 × 10²⁵; 5.98 × 10²⁴; 4.87 × 10²⁴; 6.42 × 10²³; 3.3 × 10²³; 1.31 × 10²²

Jupiter > Saturn > Neptune > Uranus> Earth > Venus > Mars > Mercury >Pluto

Area covered by water = 361,419,000 square kilometre

Area covered by land = 148,647,000 square kilometre

Ratio = 361,419,000 / 148,647,000

Ratio = 12:5

Area covered by water in standard notation = 3.61419 × 10⁸

Area covered by land in standard notation = 1.48647 × 10⁸

c = 4 – 2 + 1

c = 3

(a) For the standard form notation, only one number is kept before the decimal point and rest all numbers are written after decimal point.

The power of 10 denotes the number of terms shifted after the decimal point.

So for 9,460,000,000,000

Here only one number will be kept before decimal point that is 9.

Number of terms to be shifted after decimal point = 12

9,460,000,000,000 = 9.46 × 10¹² km

(b) One light year = 9.46 × 10¹² km

Distance between Sun and Earth = 1.496 × 10⁸ km

It is clearly seen that one light year is greater than the distance between sun and earth.

Given:

Number of sides in hexagon, n = 6

So number of diagonals = 9

1 hour = 60 minutes

So 6 hours = 60 × 6

= 360 minutes

As mentioned in the question bacteria doubles itself in every 20 minutes.

So number of times bacteria doubles itself in 360 min = 360 / 20

= 18

∴ Number of bacteria in 6 hours = 2¹⁸

Given:

Weight of blubber = 2² × 3² × 5 × 17 kg

= 4 × 9 × 5 × 17

= 36 × 85

= 3060 kg

Given:

Diameter of red blood cells = 7 × 10^–6 metres

Diameter of platelets = 2.33 × 10^–6 metre

It is clearly observed that the diameter of the red blood cell is greater than as compared to the diameter of the platelets.

(a) Given:

A googol is the number 1 followed by 100 zeroes.

So in standard notation 1 googol = 1 × 10¹⁰⁰

(b) Googol times a googol = (1 × 10¹⁰⁰) × (1 × 10¹⁰⁰)

= 10^{100 + 100}

= 10²⁰⁰

The mistake performed by the student was of multiplying the base with its exponent.