Exponents & Powers

exponent and power
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Mass of earth = 5,970,000,000,000,000,000,000,000 kg

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

Distance between Sun and Saturn = 1,433,500,000,000 m

Distance between Saturn and Uranus = 1,439,000,000,000 m.

How would we read these numbers ?

Which is heavier, Earth or Uranus?

Which distance is less ? Between Sun & Saturn or between Saturn & Uranus?

These numbers are very large and hence difficult to read, understand and compare.

To make these numbers easy to read, understand and compare, we use exponents.

Exponents

We can express a larger number as the products of its factors as

100000 = 100×1000

or 100000 = 100×50×20

or 100000 = 10×10×10×5×2×10

or 100000 = 10×10×10×10×10

In last expression, we can see that same number 10 is multiplied 5 times, we use notation 10^5 to represent the product 10×10×10×10×10.

100000 = 10×10×10×10×10 = 10^5

It is an exponential form of 100000 and is read as 10 raised to the power of 5.

Here ‘10’ is called the base and ‘5’ the exponent.

∴ An exponent refers to the number of times a number is multiplied by itself.

“Exponential form is a way of expressing a standard number using a base and a raised number called an exponent.”

➤ Numbers in expanded form can be written in exponential form

25769 = 2 × 10000 + 5 × 1000 + 7 × 100 + 6 × 10 + 9

=2\times 10^4 + 5 \times 10^3 + 7 \times 10^2 + 6 \times 10 + 9

➤ Exponential numbers having base other than 10,

16 = 2×2×2×2 = 2^4 here, 2 is the base & 4 is the exponent.
16 = 4×4 = 4^2, here 4 is the base & 2 is the exponent.

64 = 2 × 2 × 2 × 2 × 2 × 2 = 2^6, here 2 is the base & 6 is the exponent.
64 = 4 ×4 × 4 = 4^3, here 4 is the base & 3 is the exponent.

81 = 3 × 3 × 3 × 3 = 3^4, here 3 is the base & 4 is the exponent.

Some Exponents have special names

↬ Square

When the exponent is 2 i.e. when the base is raised to 2, the base is said to be squired.

3^2, which is 3 raised to the power 2, also read as 3 squared

3^2 = 3×3 =9

10^2, which is 10 raised to the power 2, also read as ‘10 squared

10^2 = 10×10 =100

↬ Cube

When the exponent is 3 i.e. when the base is raised to 3, it is said to be cubed.

2^3, which is 2 raised to the power 3, also read as 2 cubed

2^3 = 2×2×2 = 8

5^3, which is 5 raised to the power 3, also read as 5 cubed

5^3 = 5 × 5 × 5 = 125

For any integer a

a = a^1

a \times a = a^2 (read as ‘a squared’ or ‘a raised to the power 2’)

a \times a \times a = a^3 (read as ‘a cubed’ or ‘a raised to the power 3’)

a \times a \times a \times a = a^4 (read as ‘a raised to the power 4’ or the 4th power of a)

…………………………

a \times a \times a \times a \times a \times a \times a = a^7 (read as ‘a raised to the power 7’ or the 7th power of a)

and so on.

Exponential form of Negative Integer

(-2)^2= (−2)×(−2) = 4,

Value of a negative integer raised to even positive power is positive.

(-2)^3 = (−2)×(−2)×(−2) = −8,

Value of a negative integer raised to odd positive power is negative.

(-a)^m = a, for any positive even number m

(-a)^n = -a, for any positive odd number n

Exponential form as power of different factors

144 = 9×16 = 3×3×4×4 = 3^2\times 4^2

Order of factor doesn’t matter

(Multiplication is commutative )

3^2\times 4^2 = 4^2\times 3^2

a × a × a × b × b can be expressed as a^3 b^2(read as a cubed b squared)

a × a × b × b × b × b can be expressed as a^2 b^4 (read as a squared into b raised to the power of 4).

a^3 b^2 = b^2 a^3(commutative Property)

a^2 b^4 = b^4 a^2

Exponential form as power of Prime factors

(Each base must be a prime number).

144 = 9×16 =3×3×4×4 = 3×3×2×2×2×2 = 3^2 \times 2^4

Prime factorise the number first, then express it in exponential form

432 = 2 × 2 × 2 × 2 × 3 × 3 × 3

or 432 = 2^4\times 3^3

Try These

(1) Express: (i) 729 as a power of 3 (ii) 128 as a power of 2

(iii) 343 as a power of 7

(2) Express as product of powers of their prime factors:

(i) 648 (ii) 405 (iii) 540

(3) Compare the following numbers:

(i) 2\cdot 7 \times 10^{12} ; 1\cdot 5 \times 10^8 (ii) 4 \times 10^{14} ; 3 \times 10^{17}

Answer

(1) (i) 729 =3×3×3×3×3×3 = 3^6

(ii) 128 = 2×2×2×2×2×2×2 = 2^7

(iii) 343 = 7×7×7 = 7^3

(2) (i) 648 = 2×2×2×3×3×3×3 = 2^3\times 3^4

(ii) 405 = 3×3×3×3×5 = 3^4\times 5

(3) (i)2\cdot 7 \times 10^{12} > 1\cdot 5 \times 10^8

(ii) 4 \times 10^{14} < 3 \times 10^{17}

Laws of Exponents

Multiplying Powers with the same base

2^2\times 2^5 =2\times 2\times 2\times 2\times 2\times 2\times 2 = 2^7 = 2^{2+5}

(-3)^2 \times (-3)^3 = (-3)\times (-3)\times (-3)\times (-3)\times (-3)

= (-3)^5 = (-3)^{2+3}

a^2\times a^3 = a\times a\times a\times a\times a = a^5 = a^{2+3}

For any non-zero integers a, where m and n are whole numbers.

a^m\times a^n = a^{m+n}

Dividing Powers with the same base

2^5\div 2^2 =\frac {2\times 2\times 2\times 2\times2}{2\times 2}

= 2^3 = 2^{5-2}

(-3)^3\div (-3)^2 =\frac {('3)\times (-3)\times (-3)}{(-3)\times (-3)} = (-3) =(-3)^{3-2}

a^4 \div a^2 =\frac{a\times a\times a\times a}{a\times a}

= a^2 = a^{4-2}

For any non-zero integers a, where m and n are whole numbers and m > n.

a^m \div a^n = a^{m-n}

Power of a Power

(5^3)^2 = 5^3\times 5^3 = 5^{3+3} = 5^6 = 5^{3\times 2}

(3^2)^5= 3^2\times 3^2\times 3^2\times 3^2\times 3^2

= 3^{2+2+2+2+2} =3^{10} = 3^{2\times 5}

(a^2)^3 = a^2\times a^2\times a^2 =a^{2+2+2}

= a^6= a^{2\times 3}

For any non-zero integers a, where m and n are whole numbers.

(a^m)^n = a^{m×n}

Multiplying Powers with the same exponent

2^3\times 5^3 =2×2×2×5×5×5 = (2×5)×(2×5)×(2×5) = 10×10×10 = 10^3 = (2\times 5)^3

(-3)^2\times (2)^2 = (-3)×(-3)×2×2 = (-3×2)×(-3×2) = (-6)×(-6) = (-6)^2 =(-3\times 2)^2

a^2×b^2 = a×a×b×b = (a×b)×(a×b) = (ab)^2= (a\times b)^2

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

a^m\times b^m = (a\times b)^m

Dividing Powers with the same exponent

2^2 \div 3^2 = \frac{2\times 2}{3\times 3} = \frac{2^2}{3^2} = (\frac{2}{3})^2

a^4 \div b^4 = \frac {a\times a\times a\times a}{b\times b\times b\times b} = \frac{a^4}{b^4} = (\frac {a}{b})^4

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

a^m \div b^m = (\frac{a}{b})^m

Exponent Zero

2^2 \div 2^2 = \frac{2\times 2}{2\times 2} = \frac{4}{4} = 1,

also, 2^2 \div 2^2 = 2^{2-2} = 2^0 ( using laws of exponents )

2^0 = 1

5^2 \div 5^2 = \frac{5\times 5}{5\times 5} = \frac{25}{25} = 1,

also, 5^2 \div 5^2 = 5^{2-2} = 5^0( using laws of exponents )

5^0 = 1

For any non-zero integer a,

a^0 = 1

Laws of exponents

a^0 = 1 zero exponent
a^m \times a^n = a^{m+n} same base
a^m \div a^n = a^{m-n} same base
(a^m)^n = a^{m\times n} power of power
a^m \times b^m = (a\times b)^m same exponent
a^m \div b^m = (\frac{a}{b})^m same exponent

Try These

(1) Simplify and write in exponential form:

(i) 2^5 \times 2^3 (ii) p^3 \times p^2 (iii) 9^{11} \div 9^7

(iv) 20^{15} \div 20^{13} (v) (70^{15})^2 (vi) 4^3 \times 2^3

(vii) 2^5 \times b^5 (viii) 2^5 \times b^5 (ix) (-2)^3 \div b^3

(2) Simplify: \frac {(2^{5})^2\times 7^3}{8^3\times 7}

Answer

(1) (i) 2^5 + 3 = 2^8

(ii) p^3 + 2 = p^5

(iii) 9^{11-7} = 9^4

(iv) 20^{15-13} = 20^2

(v) 70^{15\times 2} = 70^{30}

(vi) (4\times 2)^3 = 8^3

(vii) (2\times b)^5= (2b)^5

(ii) (2 \div b)^5= (\frac{2}{b})^5

(iii) \left[(-2) \div b\right]^3= (\frac{-2}{b})^3

(2) \frac {(2^5)^2\times 7^3}{8^3\times 7}

= \frac{2^{5\times 2}\times 7^3}{(2^3)^3 \times 7}

= \frac{2^{10}\times 7^3}{2^9 \times 7}

= \frac{2^{10}}{2^9} \times \frac {7^3}{ 7^1}

= 2^{10-9} \times 7^{3-1}

= 2\times 7^2

= 2×49

= 98

Expansion of Integers in the Exponential form

25769 = 2 × 10000 + 5 × 1000 + 7 × 100 + 6 × 10 + 9 × 1

= 2 \times 10^4 + 5 \times 10^3 + 7 \times 10^2 + 6 \times 10^1 + 9 \times 10^0

Expansion of Decimals

Decimal numbers contain fractional part after decimal point. To expand them in exponential form, we need to know to express tenth, hundredth, thousandth etc (i.e., places after point) in exponential form. (Learn about them in full in next topic.)

Standard Index form of a number

Standard Index Form of A Number

Any number can be expressed as a decimal number between 1.0 and 10.0 including 1.0 multiplied by a power of 10 . A number in such form is called in standard form (or standard index form).

a × 10^n(where a is any real number such that 1≤|a|<10 and n is any integer).

→ |a| should be greater or equal to 1 and less than 10, i.e., −10 < a ≤ −1 and 1 ≤ a < 10.

e.g., 79 = 7\cdot 9\times 10^1

790 = 7\cdot 9\times 10^2

7900 = 7\cdot 9\times 10^3 and so on.

→ A very large number or very small number can be conveniently expressed in standard form.

e.g. 423,000,000,000,000 = 4\cdot 23\times 10^{14}

A very small number uses negative powers.

e.g. 0.000000000692 = 6\cdot 92\times 10^{-10}

Comparing numbers in Standard Index form

Now, we can answer the questions asked at beginning by expressing those large numbers into standard form,

Mass of earth = 5,970,000,000,000,000,000,000,000 kg

= 5\cdot 97\times 10^{24} kg

= five point nine seven times ten raised to the power of twenty-four.

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

= 8\cdot 68 \times 10^{25} kg

= eight point six hundred eight times ten raised to the power of twenty-five.

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

8\cdot 680\times 10^{25}\, Kg\, > \, 5\cdot 97\times 10^{24}\, Kg

∴ Uranus is heavier than earth.

Distance between Sun and Saturn = 1,433,500,000,000 m =1\cdot 4335\times 10^{12}\, m

Distance between Saturn and Uranus = 1,439,000,000,000 m.= 1\cdot 439\times 10^{12} m

Powers of 10 in both numbers are same and 1.4335 < 1.439

1\cdot 4335\times 10^{12}\, m\, < \, 1\cdot 439\times 10^{12}\,m

∴ Distance between Sun & Saturn is less then distance between Saturn and Uranus.

Try These

(1) Expand by expressing powers of 10 in the exponential form:

(i) 172 (ii) 56,439

Answer

(1)(i) 172 = 1×100 + 7×10 + 2×1 = 1\times 10^2 + 7\times 10^1 + 2\times 10^0

(ii) 56,439 = 5×10,000 + 6×1000 + 4×100 + 3×10 + 9×1

= 5\times 10^4 + 6\times 10^3 + 4\times 10^2 + 3\times10^1 + 9\times10^0

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