Laws Of Exponents

Multiplying Powers with the Same Base

(i)  Let us calculate 2² × 2³

2² × 2³ = (2 × 2) × (2 × 2 × 2)

= 2 × 2 × 2 × 2 × 2 = 2⁵ = 2²⁺³

Note that the base in 2² and 2³ 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)  a² × a⁴ = (a × a) × (a × a × a × a)

 = a × a × a × a × a × a = a⁶

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

Similarly, verify:

4² × 4² = 4²⁺²

3² × 3³ = 3²⁺³

Dividing Powers with the Same Base

Let us simplify 3⁷ ÷ 3⁴?

 

Thus 

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

Similarly,

Or

Let a be a non-zero integer, then,

Or 

Taking Power of a Power

Consider the following

Simplify ;

Now,  means 2³ is multiplied two times with itself.

(2³) ²= 2³ × 2³

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

= 2⁶ = 2^{3 × 2}

Thus

Similarly  

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

Can you tell what would( 7² )¹⁰ would be equal to?

So

 

 

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

Example 7

Can you tell which one is greater (5²) × 3 or (5²) ³ ?

 Solution

(5²) × 3 means 5² is multiplied by 3 i.e., 5 × 5 × 3 = 75

but ( 5² )³means 5² is multiplied by itself three times i.e. ,

5² × 5² × 5² = 5⁶ = 15,625

Therefore  (5²)³ > (5²) × 3

Multiplying Powers with the Same Exponents

Can you simplify 2³ × 3³? Notice that here the two terms 2³ and 3³ have different bases, but the same exponents.

Now,   2³ × 3³ = (2 × 2 × 2) × (3 × 3 × 3)

= (2 × 3) × (2 × 3) × (2 × 3)

= 6 × 6 × 6

= 6³ (Observe 6 is the product of bases 2 and 3)

Consider 4⁴ × 3⁴        = (4 × 4 × 4 × 4) × (3 × 3 × 3 × 3)

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

= 12 × 12 × 12 × 12

=12⁴

Consider, also, 3² × a²         = (3 × 3) × (a × a)

= (3 × a) × (3 × a)

= (3 × a)²

= (3a)²      (Note: 3×a = 3a )

Similarly, a⁴ × b⁴         = (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)

Example 8

Express the following terms in the exponential form:

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

Solution

(i) (2 × 3)⁵ = (2 × 3) × (2 × 3) × (2 × 3) × (2 × 3) × (2 × 3)

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

= 2⁵ × 3⁵

(ii)  (2a)⁴ = 2a × 2a × 2a × 2a

= (2 × 2 × 2 × 2) × (a × a × a × a)

= 2⁴ × a⁴

(iii)  (– 4m)³ = (– 4 × m)³

= (– 4 × m) × (– 4 × m) × (– 4 × m)

= (– 4) × (– 4) × (– 4) × (m × m × m) = (– 4)³ × (m)³

Dividing Powers with the Same Exponents

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.

Example 9

Expand:    (i)          (ii)   

Solution

(i)  

(ii)  

Numbers with exponent zero

 Can you tell what           equals to?

by using laws of exponents

 So   

Can you tell what 7⁰ 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.

Miscellaneous Examples Using The Laws Of Exponents

Let us solve some examples using rules of exponents developed.

Example 10

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

Solution

We have, 8 × 8 × 8 × 8 = 8⁴

But we know that   8 = 2 × 2 × 2 = 2³

Therefore 8⁴ = (2³)⁴ = 2³ × 2³ × 2³ × 2³

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

= 2¹²

Example 11

Simplify and write the answer in the exponential form.

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

Solution

(i)  

(ii)  

(iii) 

(iv)   6

(v)  

Therefore  

Example 12

Simplify:

(i)         (ii)    (iii) 

Solution

(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.

Decimal Number System

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 × 10⁴ + 7 × 10³ + 5 × 10² + 6 × 10¹ + 1 × 10⁰

(Note 10,000 = 10⁴, 1000 = 10³, 100 = 10², 10 = 10¹ and 1 = 10⁰)

Let us expand another number:

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

= 1 × 10⁵ + 0 × 10⁴ + 4 × 10³ + 2 × 10² + 7 × 10¹ + 8 × 10⁰

= 1 × 10⁵ + 4 × 10³ + 2 × 10² + 7 × 10¹ + 8 × 10⁰

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.

Expressing Large Numbers In The Standard From

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.

1. Sun is located 300,000,000,000,000,000,000 m from the centre of our Milky Way Galaxy.
2. Number of stars in our Galaxy is 100,000,000,000.
3. 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 × 10¹

590 = 5.9 × 100 = 5.9 × 10²

5900 = 5.9 × 1000 = 5.9 × 10³

5900 = 5.9 × 10000 = 5.9 × 10⁴    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 × 10³ is the standard form of 5,985.

Note, 5,985 can also be expressed as 59.85 × 100 or 59.85 × 10². But these are not the standard forms, of 5,985. Similarly, 5,985 = 0.5985 × 10,000 = 0.5985 × 10⁴ 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 × 10²⁰ 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 × 10¹⁰

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

= 5.976 × 10²⁴ 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 × 10²⁵ 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 × 10¹² m.

The distance betwen Saturn and Uranus is 1,439,000,000,000 m or 1.439 × 10¹² m.

The distance between Sun and Earth is 149, 600,000,000 m or 1.496 × 10¹¹m.

Can you tell which of the three distances is smallest?

Example 13

Express the following numbers in the standard form:

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

Solution

(i)   5985.3 = 5.9853 × 1000 = 5.9853 × 10³

(ii)   65,950 = 6.595 × 10,000 = 6.595 × 10⁴

(iii)  3,430,000 = 3.43 × 1,000,000 = 3.43 × 10⁶

(iv)  70,040,000,000 = 7.004 × 10,000,000,000 = 7.004 × 10¹⁰

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.