Laws of Exponents

(i) Let us calculate 2^{2} × 2^{3}

2^{2} × 2^{3} = (2 × 2) × (2 × 2 × 2)

= 2 × 2 × 2 × 2 × 2 = 2^{5} = 2^{2+3}

Note that the base in 2^{2 }and 2^{3} is same and the sum of the exponents, i.e., 2 and 3 is 5

(ii) (–3)^{4 }× (–3)^{3} = [(–3) × (–3) × (–3)× (–3)] × [(–3) × (–3) × (–3)]

= (–3) × (–3) × (–3) × (–3) × (–3) × (–3) × (–3)

= (–3)^{7}

= (–3)^{4+3}

Again, note that the base is same and the sum of exponents, i.e., 4 and 3, is 7

(iii) a^{2} × a^{4} = (a × a) × (a × a × a × a)

= a × a × a × a × a × a = a^{6}

(Note: the base is the same and the sum of the exponents is 2 + 4 = 6)

Similarly, verify:

4^{2 }× 4^{2} = 4^{2+2}

3^{2} × 3^{3} = 3^{2+3}

Let us simplify 3^{7} ÷ 3^{4}?

Thus

(Note, in 37 and 34 the base is same and 3^{7} ÷ 3^{4} becomes 3^{7–4})

Similarly,

Or

Let a be a non-zero integer, then,

Or

Consider the following

Simplify ;

Now, means 2^{3} is multiplied two times with itself.

(2^{3}) ^{2}= 2^{3} × 2^{3}

= 2^{3 + 3} (Since a^{m} × a^{n} = a^{m + n})

= 2^{6 }= 2^{3 × 2}

Thus

Similarly

(Observe 8 is the product of 2 and 4).

Can you tell what would( 7^{2} )^{10} would be equal to?

So

^{
}

From this we can generalise for any non-zero integer ‘a’, where ‘m’ and ‘n’ are whole numbers,

Can you tell which one is greater (5^{2}) × 3 or (5^{2}) ^{3} ?

(5^{2}) × 3 means 5^{2} is multiplied by 3 i.e., 5 × 5 × 3 = 75

but ( 5^{2 })^{3}means 5^{2} is multiplied by itself three times i.e. ,

5^{2} × 5^{2} × 5^{2} = 5^{6} = 15,625

Therefore (5^{2})^{3} > (5^{2}) × 3

Can you simplify 2^{3} × 3^{3}? Notice that here the two terms 2^{3} and 3^{3} have different bases, but the same exponents.

Now, 2^{3} × 3^{3} = (2 × 2 × 2) × (3 × 3 × 3)

= (2 × 3) × (2 × 3) × (2 × 3)

= 6 × 6 × 6

= 6^{3} (Observe 6 is the product of bases 2 and 3)

Consider 4^{4} × 3^{4} = (4 × 4 × 4 × 4) × (3 × 3 × 3 × 3)

= (4 × 3) × (4 × 3) × (4 × 3) × (4 × 3)

= 12 × 12 × 12 × 12

=12^{4}

Consider, also, 3^{2} × a^{2 } = (3 × 3) × (a × a)

= (3 × a) × (3 × a)

= (3 × a)^{2}

= (3a)^{2} (Note: 3×a = 3a )

Similarly, a^{4} × b^{4} = (a × a × a × a) × (b × b × b × b)

= (a × b) × (a × b) × (a × b) × (a × b)

= (a × b)^{4}

= (ab)^{4} (Note a × b = ab)

In general, for any non-zero integer a

a^{m} × b^{m} = (ab)^{m} (where m is any whole number)

Express the following terms in the exponential form:

(i) (2 × 3)^{5 }(ii) (2a)^{4} (iii) (– 4m)^{3}

**(i)** (2 × 3)^{5} = (2 × 3) × (2 × 3) × (2 × 3) × (2 × 3) × (2 × 3)

= (2 × 2 × 2 × 2 × 2) × (3 × 3× 3 × 3 × 3)

= 2^{5} × 3^{5}

**(ii)** (2a)^{4} = 2a × 2a × 2a × 2a

= (2 × 2 × 2 × 2) × (a × a × a × a)

= 2^{4} × a^{4}

**(iii)** (– 4m)^{3} = (– 4 × m)^{3 }

= (– 4 × m) × (– 4 × m) × (– 4 × m)

= (– 4) × (– 4) × (– 4) × (m × m × m) = (– 4)^{3} × (m)^{3}

Observe the following simplifications:

From these examples we may generalise

where a and b are any non-zero integers and m is a whole number.

Expand: (i) (ii)

(i)

(ii)

Can you tell what equals to?

by using laws of exponents

So

Can you tell what 7^{0 }is equal to?

And

Therefore

Similarly

And

Thus (for any non-zero integer a)

So, we can say that any number (except 0) raised to the power (or exponent) 0 is 1.

Let us solve some examples using rules of exponents developed.

Write exponential form for 8 × 8 × 8 × 8 taking base as 2.

We have, 8 × 8 × 8 × 8 = 8^{4}

But we know that 8 = 2 × 2 × 2 = 2^{3}

Therefore 8^{4} = (2^{3})^{4} = 2^{3} × 2^{3} × 2^{3} × 2^{3}

= 2^{3 × 4} [You may also use (a^{m})^{n} = a^{mn}]

= 2^{12}

Simplify and write the answer in the exponential form.

(i) (ii) (iii) (iv) 3 (v)

(i)

(ii)

(iii)

(iv) 6

(v)

Therefore

Simplify:

(i) (ii) (iii)

(i) We have

(ii)

(iii)

**Note**: In most of the examples that we have taken in this Chapter, the base of a power was taken an integer. But all the results of the chapter apply equally well to a base which is a rational number.

Let us look at the expansion of 47561, which we already know:

47561 = 4 × 10000 + 7 × 1000 + 5 × 100 + 6 × 10 + 1

We can express it using powers of 10 in the exponent form:

Therefore, 47561 = 4 × 10^{4} + 7 × 10^{3} + 5 × 10^{2} + 6 × 10^{1} + 1 × 10^{0}

(Note 10,000 = 10^{4}, 1000 = 10^{3}, 100 = 10^{2}, 10 = 10^{1} and 1 = 10^{0})

Let us expand another number:

104278 = 1 × 100,000 + 0 × 10,000 + 4 × 1000 + 2 × 100 + 7 × 10 + 8 × 1

= 1 × 10^{5 }+ 0 × 10^{4} + 4 × 10^{3} + 2 × 10^{2} + 7 × 10^{1} + 8 × 10^{0}

= 1 × 10^{5} + 4 × 10^{3} + 2 × 10^{2} + 7 × 10^{1} + 8 × 10^{0}

Notice how the exponents of 10 start from a maximum value of 5 and go on decreasing by 1 at a step from the left to the right upto 0.

Let us now go back to the beginning of the chapter. We said that large numbers can be conveniently expressed using exponents. We have not as yet shown this. We shall do so now.

- Sun is located 300,000,000,000,000,000,000 m from the centre of our Milky Way Galaxy.
- Number of stars in our Galaxy is 100,000,000,000.
- Mass of the Earth is 5,976,000,000,000,000,000,000,000 kg.

These numbers are not convenient to write and read. To make it convenient we use powers.

Observe the following:

59 = 5.9 × 10 = 5.9 × 10^{1}

590 = 5.9 × 100 = 5.9 × 10^{2}

5900 = 5.9 × 1000 = 5.9 × 10^{3}

5900 = 5.9 × 10000 = 5.9 × 10^{4} and so on

We have expressed all these numbers in the **standard form**. 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 a form of a number is called its **standard form**. Thus,

5,985 = 5.985 × 1,000 = 5.985 × 10^{3 }is the standard form of 5,985.

Note, 5,985 can also be expressed as 59.85 × 100 or 59.85 × 10^{2}. But these are not the standard forms, of 5,985. Similarly, 5,985 = 0.5985 × 10,000 = 0.5985 × 10^{4} is also not the standard form of 5,985.

We are now ready to express the large numbers we came across at the beginning of the chapter in this form.

The, distance of Sun from the centre of our Galaxy i.e.,

300,000,000,000,000,000,000 m can be written as.

3.0 × 100,000,000,000,000,000,000 = 3.0 × 10^{20} m

Now, can you express 40,000,000,000 in the similar way?

Count the number of zeros in it. It is 10.

So, 40,000,000,000 = 4.0 × 10^{10}

Mass of the Earth = 5,976,000,000,000,000,000,000,000 kg

= 5.976 × 10^{24} kg

e fact, that the number when written in the standard form is much easier to read, understand and compare than when the number is written with 25 digits?

Now,

Mass of Uranus = 86,800,000,000,000,000,000,000,000 kg

= 8.68 × 10^{25} kg

Simply by comparing the powers of 10 in the above two, you can tell that the mass of Uranus is greater than that of the Earth.

The distance between Sun and Saturn is 1,433,500,000,000 m or 1.4335 × 10^{12} m.

The distance betwen Saturn and Uranus is 1,439,000,000,000 m or 1.439 × 10^{12} m.

The distance between Sun and Earth is 149, 600,000,000 m or 1.496 × 10^{11}m.

Can you tell which of the three distances is smallest?

Express the following numbers in the standard form:

(i) 5985.3 (ii) 65,950 (iii) 3,430,000 (iv) 70,040,000,000

(i) 5985.3 = 5.9853 × 1000 = 5.9853 × 10^{3}

(ii) 65,950 = 6.595 × 10,000 = 6.595 × 10^{4}

(iii) 3,430,000 = 3.43 × 1,000,000 = 3.43 × 10^{6}

(iv) 70,040,000,000 = 7.004 × 10,000,000,000 = 7.004 × 10^{10}

A point to remember is that one less than the digit count (number of digits) to the left of the decimal point in a given number is the exponent of 10 in the standard form. Thus, in 70,040,000,000 there is no decimal point shown; we assume it to be at the (right) end. From there, the count of the places (digits) to the left is 11. The exponent of 10 in the standard form is 11 – 1 = 10. In 5985.3 there are 4 digits to the left of the decimal point and hence the exponent of 10 in the standard form is 4 – 1 = 3.

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