Negative Exponents

negative exponent
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Thickness of a piece of paper is 0.00016 m

The size of a plant cell is 0.00001275 m.

Diameter of a wire on a computer chip is 0.000003 m.

The average diameter of a Red Blood Cell is 0.000007 m.

Arrange these in ascending order of size ?

We know how to write large numbers more conveniently using exponents. Can we write these very small numbers in exponential form?

Powers with Negative Exponent

10^2 = 10\times 10 =100\\  \ 10^1 = 10\\  \ 10^0 = 1\\  \ 10^{-1} = ?

As the exponent decreases by 1, the value becomes one-tenth of the previous value.

Continuing the above pattern we get,

10^{−1} = \frac {1}{10}\\  \ 10^{-2} = \frac {1}{10} \div 10 = \frac {1}{100} = \frac {1}{10^2}\\  \ 10^{−3} = \frac {1}{100} \div 10 = \frac {1}{1000} =\frac {1}{10^3}

Let’s change the base and observe the pattern,

3^2 = 3\times 3 = 9\\  \ 3^1 = 3\\  \ 3^0 = 1\\  \ 3^{-1} = ?

As the exponent decreases by 1, the value becomes one-third of the previous value.

Continuing the above pattern we get,

3^{-1} = \frac {1}{3}\\  \ 3^{-2} = \frac {1}{3} \div 3 = \frac {1}{3\times 3} =\frac {1}{3^2}\\  \ 3^{-3} = \frac {1}{3\times 3} \div 3 = \frac {1}{3\times 3\times 3} =\frac {1}{3^3}

↬ In general, we can say that, for any non-zero integer a and m,

a^{-m} =\frac {1}{a^m}

We know that \frac{1}{a^m} is multiplicative inverse of a^m

a^{-m} is the multiplicative inverse (reciprocal) of a^m.

↬ We can expand decimals using negative exponents as,

526.193 = 5 × 100 + 2 × 10 + 6 × 1 + 1/10 + 9/100 + 3/1000

= 5 \times 10^2 + 2 \times 10 + 6 \times 1 + \frac {1}{10} + \frac {9}{10^2} + \frac {1}{10^3}

= 5 \times 10^2 + 2 \times 10^1 + 6 \times 10^0 + 1\times 10^{-1} + 9\times 10^{-2} + 3\times 10^{-3}

Laws of Exponents

Laws of exponents which hold for positive exponents also hold for negative exponents.

For any non-zero integers a and b and any integers m and n .

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
(\frac{a}{b})^{-m}=(\frac{b}{a})^m negative exponent

Expressing very small number in Standard Index 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 (or standard index form).

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

e.g.,

799 = 7.99\times 10^2\\  \79.9 = 7.9\times 10\\  \0.799 = 7.99\times 10^{-1}\\  \0.0799 =7.99\times 10^{-2}\\  \0.00799 =7.99\timed 10^{-3} & so on.

Very small numbers can be expressed in standard form using negative exponents.

→ Now we can express small number asked earlier in standard form

Thickness of a piece of paper is 0.000016\, m\, = 1.6\times 10^{-5}\, m

The size of a plant cell is 0.00001275\, m\, = 1.275\times 10^{-5}\, m

Diameter of a wire on a computer chip is 0.000003 \,m\, = 3\times 10^{-6}\, m .

The average diameter of a Red Blood Cell is 0.000000007\, m = 1.6\times 10^{-9}\, m.

Comparing numbers in Standard Index Form

To compare numbers in standard form, we convert them into numbers with the same exponents.

❔ Diameter of the Sun = 1.4 \times 10^9\, m

Diameter of the earth =1.2756 \times 10^7 m

Which is bigger and by how many times ?

↬ Diameter of the Sun = 1.4 \times 10^9\, m

= 1.47\times 10^2\times 10^7 = 147\times 10^7\, m

Diameter of the Earth = 1.2756 \times 10^7 m

Now, the exponents are equal, we can compare them, 147 > 1.2756

So, Sun is bigger than Earth by =\frac {147 \times 10^7 }{1.2756 \times 10^7} ≅ 100 times.

Adding/Subtracting numbers in Standard Form

○ To add or subtract numbers in standard form, we convert them into numbers with the same exponents.

e.g.,

1.496 \times 10^{11} - 3.84 \times 10^8\\  \ = 1496 \times 10^8 - 3.84 \times 10^8\\  \ = (1496 - 3.84) \times 10^8\\  \ = 1492.16 \times 10^8\\  \ = 1.49216 \times 10^{11}(In standard form)

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