Algebra and Powers. The Five Fundamental Properties

The use of powers in algebra is very widespread. In this page we will try to know and understand their fundamental properties .

We talk about powers when we are dealing with multiplications of equal factors. A great way to simplify: for example, instead of writing 4x4x4x4x4 (multiplication between 5 equal factors), we could write this operation in another way, that is with 4 raised to the fifth power (4⁵).

In powers we have a base (real number, therefore also negative) and an exponent (n, an integer). The result is obtained by multiplying the base by itself, as many times as those indicated by the exponent.

After this brief parenthesis (to delve deeper, or review the topic, read ” Powers, what they are and how to calculate them “) let’s move on to the five rules.

Table of Contents

The five fundamental properties of powers

As anticipated several times, there are five main rules that characterize powers. Let’s list them below, trying to explain them one by one with some examples, of the exercises done.

Summary

1) The power of a product is equal to the product of the powers of its factors

In general: (a b c)ⁿ = aⁿ bⁿ cⁿ
where a, b and c are the bases and “ⁿ” the exponent.

What does it mean?

This property teaches us that if I have a multiplication between several different factors, raised to a power ⁿ, I can take the exponent on each factor and multiply the powers together.

If it is not clear yet, it will certainly be clear by doing some numerical exercises.

First exercise
(2 3 5)² = 2² 3² 5²
in fact, in the first case (to the left of the equals sign) we will have 2 3 5 = 30, therefore 30² = 30 30 = 900
while on the right we will have 2² 3² 5² = 4 9 25 = 36 25 = 900
900 = 900
Therefore, the initial equality is correct. The property has been proven.
Second exercise
(4·1·5)⁴ = 4⁴·1⁴·5⁴
solving on the left and on the right, we will have:
on the left: (4·1·5)⁴ = (20)⁴ = 160,000
on the right: 4⁴·1⁴·5⁴ = 256·1·625 = 160,000
Perfect!

2) The power of a quota (or quotient) is equal to the quota (or quotient) of the powers of the dividend and the divisor.

Premise (difference between “quotient and quota”)
When you do divisions, the result is called “quotient” when it is a whole number and without a remainder, or without decimals; when instead the result you get includes a remainder or a decimal number, then it is called “quotient”.

For example, 15 divided by 5 gives 3 as a result, without a remainder or decimals. The “3” is called “quotient”.
Differently, 16 divided by 5 gives “3 with a remainder of 1”, or, in decimals, “3.2”. In this case we have a “quotient”.

Having closed the premise, let’s move on to the rule.

In general: (a/b)ⁿ = aⁿ/bⁿ

What does it mean?

We have that (“a” divided by “b”) raised to the power of “n” is equal to (“a” raised to the power of “n”) divided by (“b” raised to the power of “n”).

Here too, as for the first property, we have distributed the exponent by removing the parenthesis (in fact, we speak of the distributive property in the first two rules).

A division (or fraction) raised to the power is therefore equal to the division between the powers of the fraction (dividend and divisor).

Let’s try with two examples.

First exercise
(4/2)³ = 4³/2³
on the left: (4/2)³ = (2)³ = 8
on the right: 4³/2³ = 64/8 = 8
Second exercise
(11/4)⁴ = 11⁴/4⁴
on the left: (11/4)⁴ = (2.75)⁴ = 57.19
on the right: 11⁴/4⁴ = 14.641/256 = 57.19

From the first two rules we learned that, for multiplications and divisions, the distributive property comes into play. This will not be the case, as we will see, for additions and subtractions.

3) The product of powers with the same base is a power having the same base and the sum of the exponents as its exponent.

In general, aⁿ aᵐ aᵖ = a⁽ⁿ ⁺ ᵐ ⁺ ᵖ⁾
where “a” to the power of “n” times “a” to the power of “m” times “a” to the power of “p” is equal to “a” to the power of “(n + m + p)”.

What does this mean?

When we are faced with the multiplication of multiple powers with the same base and different exponents, we can safely rewrite the operation using the base (“a”) only once and inserting the sum of the exponents as the exponent of that base.

As usual, it will be easier with some exercises.

First exercise
2³ 2⁴ 2² = 2⁽³⁺⁴⁺²⁾
on the left: 2³ 2⁴ 2² = 8 16 4 = 512
on the right: 2⁽³⁺⁴⁺²⁾ = 2⁹ = 512
Second exercise
5³ 5⁷ 5⁵ 5⁴ = 5⁽³⁺⁷⁺⁵⁺⁴⁾ = 5¹⁹

Simple enough, right?
Multiplication, same base, exponents add.

4) The power of a power has as its exponent the product of the exponents.

In general, (aⁿ)ᵐ = a⁽ⁿ・ᵐ⁾

If before, in the multiplication between powers with the same base we added the exponents, now, in the case of powers of powers we must multiply them.

Let’s start immediately with the examples.

First exercise
(2³)⁴ = 2⁽³・⁴⁾ = 2¹²
on the left: (2³)⁴ = 8⁴ = 4.096
on the right: 2¹² = 4.096
Second exercise
(5³)⁵ = 5⁽³・⁵⁾ = 5¹⁵
Third exercise
[(2³)⁴]² = 2⁽³・⁴・²⁾ = 2²⁴

Clear, right?

Let’s move on to the last property.

5) The quotient of two powers with the same base has as its exponent the difference of the exponents.

We talk about quotient because, as explained at the beginning, we have an integer result (without remainder or decimals), since the bases are always equal. Otherwise we would have talked about quotient.

In general, aⁿ/aᵐ = a⁽ⁿ ⁻ ᵐ⁾

Just remember that, for multiplications between powers with the same base, the exponents were added, while now, in divisions, they are subtracted.

So, exercises.

First exercise
2⁴/2² = 2⁽⁴⁻²⁾
left: 2⁴/2² = 16/4 = 4
right: 2⁽⁴⁻²⁾ = 2² = 4
Second exercise
5⁷/5⁴ = 5⁽⁷⁻⁴⁾ = 5³
Third exercise
2²/2⁴ = 2⁽²⁻⁴⁾ = 2⁻² = 1/(2²) = 1/4
let’s remember that a power with a negative exponent is equal to the inverse of the base with a positive exponent to the divisor