Exponents

Earlier you had learnt that adding the same number over and over again is the same thing as multiplication.

For example;

\( \displaystyle 4+4+4+4+4+4+4 \)

instead of this equation we can use multiplication.  Because we are adding the number 4, 7 times:

 \( \displaystyle 4+4+4+4+4+4+4=7.4 \)

Now, we are going to focus on multiplying the same number several times. We are going to develop methods to make this easy for us as well as to show this in a simple way.



For example ;

\( \displaystyle 4+4+4+4+4+4+4 \)

Or if it has further steps even;

\( \displaystyle 4+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4.... \)

Writing the equation down like this every time is hard isn’t it? Can’t we find a solution that shows this equation in a simpler way?

We write the number that the initial number will be multiplied by on the right top corner, in a neat manner,  we call these Exponents. Exponents are also called Powers or Indices.

or \( \displaystyle 4^{25} \) shows that 4 must be multiplied by itself 25 times .

Base and Power

\( \displaystyle 4 \) is base ..

\( \displaystyle 7 \) is exponent , index , power

Now let us write down some exponents and their values. 

Form	Base	Power (index)	Value Of number
\( \displaystyle 5^3 \)	\( \displaystyle 5 \)	\( \displaystyle 3 \)	\( \displaystyle 5.5.5=125 \)
\( \displaystyle 4^6 \)	\( \displaystyle 4 \)	\( \displaystyle 6 \)	\( \displaystyle 4.4.4.4.4.4=4096 \)
\( \displaystyle 2^5 \)	\( \displaystyle 2 \)	\( \displaystyle 5 \)	\( \displaystyle 2.2.2.2.2=32 \)
\( \displaystyle 3^2 \)	\( \displaystyle 3 \)	\( \displaystyle 2 \)	\( \displaystyle 3.3=9 \)
\( \displaystyle 2^3 \)	\( \displaystyle 2 \)	\( \displaystyle 3 \)	\( \displaystyle 2.2.2=8 \)
\( \displaystyle 5^1 \)	\( \displaystyle 5 \)	\( \displaystyle 1 \)	\( \displaystyle 5 \)
\( \displaystyle 1^6 \)	\( \displaystyle 1 \)	\( \displaystyle 6 \)	\( \displaystyle 1.1.1.1.1.1=1 \)
\( \displaystyle 0^5 \)	\( \displaystyle 0 \)	\( \displaystyle 5 \)	\( \displaystyle 0.0.0.0.0=0 \)

Please be careful!

The most common mistake with exponents;

You are multiplying two numbers without thinking. However, the meaning of this is that the two base numbers must be multiplied by each other.

\( \displaystyle 5^2=5.5=25 \)

How to read exponents ?

All general ways of reading exponential numbers are found below;

\( \displaystyle 6^4 \)

* 6 to the 4th

* 6 to the power of 4

Only when the exponent (power ) is 2 or 3 do we have a special way of reading.

\( \displaystyle 7^2 \)

7 to the 2nd

7 to the power of 2

7 squared *(Preferred )

\( \displaystyle 5^3 \)

5 to the 3rd

5 to the power of 3

5 cubed *(Preferred)

Why are there special ways to read squared and cubed numbers?

If the exponent number 2 >> squared

if the exponent number is 3 >> cubed

Why do we read these numbers in this way?

Why do we say ‘squared’ for multiplying a number by itself?

As the name suggests, squaring something means to turn it into a square.

While finding the area of a square (how many squares are inside the square), we multiply both sides with one another.

There are 5 squares at the bottom row, and there are 5 rows.

5 units . 5 units= 25 unit²

25 unit squares... there are 25 squares within the square. Count if you like...

What you must be careful about is multiplying the number with itself once is called the square of a number.

5 units * 5 units = 25 units squared

Why do we call it the cube of a number?

To find the volume of a cube, we multiply each side with one another so we have “cubed” it.

For example, the volume of a cube with a 3-unit side is: 3 units. 3 units. 3 units = 27 unit³

If you count, you will see there are 27 cubes.