What is 3 to the 2nd power mean?

If we want to show a very large number of very small numbers in a simple manner then exponents and power come into consideration, for example, if we have to show 5 × 5 × 5 × 5 in an easy way, then we can write it as 5⁴, where 4 is the exponent and 5 is the base. The whole expression 5⁴ is said to be power. 

We can expand any power as:

2³ (read as 2 raised to power 3) = 2 × 2 × 2 

5² (5 raised to power 2) = 5 × 5

What are Exponents?

An exponent of a number is defined as the number of times the number is multiplied by itself. If 3 is multiplied by itself for n number of times, then, it is shown as:

3 × 3 × 3 × 3 × …..n times = 3ⁿ

 3ⁿ, is said as 3 raised to the power n. Therefore, exponents are also called power.

Some examples are as: 

125 = 5 × 5 × 5 = 5³

16 = 2 × 2 × 2 × 2 = 2⁴

512 = 8 × 8 × 8 = 8³

Generally exponent can be expressed as any number ‘y’ raised to power ‘n’ expressed as:

yⁿ = y × y × y × y ×………n times

 n is the nth power of a

‘y’ is the base and ‘n’ is the exponent or index or power.

‘y’ is multiplied ‘n’ times.

Laws of Exponents

Multiplication Law

The multiplication law of exponents, when two numbers having the same base raise to some different power the result is the number raised to the sum of the power of exponents.

y^m × yⁿ  = y^m+n

Example: 2⁴ × 2² = 2⁴⁺² = 2⁶ 

                3² × 3³ = 3²⁺³ = 3⁵

Division Law

When two exponents having the same bases and different powers are divided, then the result is base raised to the power difference between two powers.

a^m ÷ aⁿ  = a^m / aⁿ  = a^m-n

Example: 5⁶ / 5³ = 5^6-3 = 5³

                4⁴ / 4² = 4^4-2 = 4²

If  any of the base having negative power, then its result is its reciprocal but with positive power or integer to the base.

a^-x = 1/a^x

Some of the common rules of exponent

Rule 1: a⁰ = 1

Any number as base raised to the power 0 is always 1.

Example: 5⁰ = 1

              -4⁰ = 1

Rule 2: (a^m)ⁿ = a^(mn)

A number let say ‘a’ raised to the power ‘m’ raised to the power ‘n’ is equal to ‘a’ raised to the power product of ‘m’ and ‘n’.

Example: (5²)³ = 5^{2 × 3} = 5⁶

                (3³)³ = 3^{3 × 3} = 3⁹

Rule 3: a^m × b^m = (ab)^m

If the power of two numbers is the same with a different base then their result will be the product of the base to the power of the exponent.

Example: 5² × 6² = (5 × 6)²

                     3⁴ × 5⁴ = (3 × 5)⁴

Rule 4: a^m/b^m = (a/b)^m

The division of two numbers raised to the same powers then their result will be the whole division to the power of the exponent.

Example: 5²/6² = (5/6)²

            13⁴/15⁴ = (13/15)⁴

What is 3 to the 2nd power mean?

Solution:

Given that the number is 3 and the power is 2.

Write the number with its exponent.

3² = 3 × 3

3² = 9

Hence, the value of 3 to the 2nd power is 9.

Applications of Exponents and Powers 

Scientific notation uses the power of ten expressed as exponents, let us take an example and clear view.

The distance between two universal bodies is very-very large, take the distance between the Sun and the Earth is 149,600,000 km. If we take mass of  the Sun is 1,989,000,000,000,000,000,000,000,000,000 kg. The age of the Earth is 4,550,000,000 years. On the other side if we take mass of electron is 0.000091093837015(28). These numbers are very large or very small to memorize. With the help of exponents and powers, these huge numbers can be reduced to a very short form and can be easily written in powers of 10.

With the help of exponents and power, we can write all above big or small numbers as:

Distance between the Sun and the Earth 149,600,000 = 1.496 ×10 × 10 × 10 × 10 × 10× 10 × 10 = 1.496× 10⁸ km.

Mass of the Sun: 1,989,000,000,000,000,000,000,000,000,000 kilograms = 1.989 × 10³⁰ kilograms.

Age of the Earth:  4,550,000,000 years = 4.55 × 10⁹ years

Mass of electron:  0.000091093837015(28) = 9.1093 × 10^-31

Sample Questions

Question 1: Write 2 × 2 × 2 × 2 × 2 × 2 × 2 in exponent form.

Solution:

In this problem 2s are written 7 times, so the problem can be rewritten as an exponent of 7.

 2 × 2 × 2 ×  2 × 2 × 2 × 2 = 2⁷.

Question 2: Simplify 15³/5³  

Solution:  

Using Law: a^m/b^m = (a/b)^m

15³/5³  can be written as (15/5)³ = 3³ = 9.

Question 3: Find the value of 5^-3 * 1/5³.

Solution:

5^-3 can be written as 1/5³

so we have (1/5³)*(1/5³) = (1/5)^{3 × 3}

                                               = (1/5)⁹

Question 4: Evaluate (√4)^-3

Solution:

(√4)^-3 = (4^½)^-3

= 4^-3/2 = 1/4^3/2

= 1/(4³)^½ = 1/(64)½

= 1/(8²)½ = 1/8

Question 5: If a infant baby bear weighs 3 kg, calculate how many kgs a four-year-old bear weigh if its weight increases by the power of 2 in 4years?

Solution:

Given,

Weight of new-born bear = 3 kg

Rate of weight increase in 5 years is power of 2

Thus, the weight of the 5-year old bear = 3² = 9 kg

Question 6: Express 0.000000008537 in standard form.

Solution:

  0.000000008537

= 0.000000008537 × 10¹⁰ / 10¹⁰

= 8.537 ×10^-10

Question 7: Find x, (11/9)³ × (9/11)⁶ = (11/9)^2x-4

Solution:

Given:

(11/9)³ × (9/11)⁶ = (11/9)^2x-4

the equation can be written as:

(11/9)³ × (11/9)^-6 = (11/9)^2x-1

⇒ (11/9)^3-6 = (11/9)^2x-4

By comparing their bases we have, -3 = 2x – 4

2x = -3 + 4

x = 1/2

x = 1/2