Exponents and roots

πŸ†Practice powers and roots

Exponents and Roots

What is an exponent?

Exponentiation is the requirement for the number to be multiplied by itself several times.

What is a root?

A root is the inverse operation of exponentiation, which helps us discover which number multiplied by itself gives this result.

The square root is equal to the power of 0.5.

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\( 11^2= \)

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Powers and roots

What is an exponent?

Exponentiation is the requirement for a number to be multiplied by itself several times.
In other words, when we see a number raised to a certain power, we know that we need to multiply the number by itself several times to reach the actual number.

How do you correctly read a number with an exponent?

Let's learn through an example:
*Notations in a Word file*

Exponent base – The exponent base is the number that is required to be multiplied by itself a certain number of times.
How do we identify it?
The main number written in large – in our example, this number is 44.

Exponent – The exponent is the number that determines how many times the base is required to be multiplied by itself.
How do we identify it?
The exponent is written in small and appears to the right above the base – in our example, this number is 22.

We read it like this: 44 to the power of 22.

How do you solve an exponent?

To solve an exponent, we need to multiply the base of the exponent by itself the number of times the exponent requires us to.
Let's return to the example:

*Symbols in Word file*

Base of the exponent = 44
Exponent = 22

Let's take the base of the exponent and multiply it by itself 22 times.
We get:
4βˆ—4=164*4=16
and actually -
42=164^2=16

Let's practice another example.

How do you solve the following exponentiation:
53=5^3=

Solution:
Let's understand what the base and the exponent are.
Base of the exponent = 55
Exponent = 33
This means we need to multiply 55 by itself 33 times.
We get:
5βˆ—5βˆ—5=1255*5*5=125
And actually:
53=1255^3=125

Another example:

Solve the following power:
33=3^3=

Solution:
At first glance, we see that the base of the exponent and the exponent itself are identical. Does this change anything for us? Not at all, we work according to the rules.
We multiply the number 33 by itself – for 33 times and get:
3βˆ—3βˆ—3=273*3*3=27

And actually –
33=273^3=27

Another example:
Solve the following power:
14=1^4=
We need to take the number 11 and multiply it by itself 44 times. We get:
1βˆ—1βˆ—1βˆ—1=11*1*1*1=1
What would happen if we had such a power?
1700=1^{700}=
Would we really need to write the number 11 for 700700 times to understand that the result will be 11 in the end?
No.
From this, we can conclude that: 11 to the power of any number equals 11.


Point to Ponder – What happens when the exponent is 11?
When the exponent is 11, the number does not change at all and it can be considered as if it has already performed the exponentiation.
For example:
71=77^1=7
Any number to the power of 11 is the number itself.

Another point to consider – what happens when the exponent is 00?
When the exponent of the number is 00, we get a result of 11. It doesn't matter what the number is.
Any number to the power of 00 will equal 11.
That means:
20=12^0=1
​​​​​​​70=1​​​​​​​7^0=1
4,6750=1{4,675}^0=1

What is a root?

A root is equal to the power of 0.50.5 and is denoted by the symbol √√.
We can say that: a=a0.5\sqrt{a}=a^{0.5}
A root is the inverse operation of exponentiation.
If a small number appears on the left side, it will be the order of the root.

When any number appears as a regular root, we ask ourselves which number we would need to multiply by itself only twice to get the number inside the root?
In other words, which number raised to the power of 22 will give us the number that appears inside the root.
For example:
4=2\sqrt4=2

If we multiply 22 by itself twice, we get 44.

Another example:
16=\sqrt16=

Solution:
If we multiply the number 44 twice by itself, we get 44 and therefore:
16=4\sqrt16=4

What should you know about roots?

  • The result of the square root will always be positive!
    You will never get a negative result. We can get a result of 00.
  • There is no answer for negativeΒ number\sqrt{negative~number}!

It is important to know - roots and exponents take precedence over all four arithmetic operations.
First, we perform the root and exponentiation operations, and only then do we proceed to the order of arithmetic operations.

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