## Negative Exponents

 Image Credit: Pixabay.com

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

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

Continuing the above pattern we get,

Let’s change the base and observe the pattern,

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

Continuing the above pattern we get,

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

We know that is multiplicative inverse of

is the multiplicative inverse (reciprocal) of .

↬ We can expand decimals using negative exponents as,

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

## 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 .

 → = 1 zero exponent → same base → same base → power of power → same exponent → same exponent → 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).

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

e.g.,

& 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

The size of a plant cell is

Diameter of a wire on a computer chip is .

The average diameter of a Red Blood Cell is .

## 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 =

Diameter of the earth =

Which is bigger and by how many times ?

↬ Diameter of the Sun

Diameter of the Earth =

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

So, Sun is bigger than Earth by = ≅ 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.,

(In standard form)