The laws of exponents and radicals establish a simplified or summarized way of working on a series of numerical operations with powerswhich follow a set of mathematical rules.
For its part, the expression an is called power, (a) represents the base number and (not nth) is the exponent that indicates how many times the base must be multiplied or raised according to what is expressed in the exponent.
Laws of exponents
The purpose of the laws of exponents is to summarize a numerical expression that, if expressed in a complete and detailed manner, would be very extensive. For this reason, in many mathematical expressions they are expressed as powers.
Examples:
52 is the same as (5) ∙ (5) = 25. That is, 5 must be multiplied twice.
23 is the same as (2) ∙ (2) ∙ (2) = 8. That is, you must multiply 2 three times.
This way, the numerical expression is simpler and less confusing to solve.
1. Power with exponent 0
Any number raised to an exponent 0 is equal to 1. It should be noted that the base must always be different from 0, that is, a ≠ 0.
Examples:
a0 = 1
-50 = 1
2. Power with exponent 1
Any number raised to an exponent of 1 is equal to itself.
Examples:
a1 = a
71 = 7
3. Product of powers of equal base or multiplication of powers of equal base
What happens if we have two equal bases (a) with different exponents (n)? That is, an ∙ am. In this case, the equal bases are maintained and their powers are added, that is: an ∙ am = an+m.
Examples:
22 ∙ 24 is the same as (2) ∙ (2) x (2) ∙ (2) ∙ (2) ∙ (2). That is, add the exponents 22+4 and the result would be 26 = 64.
35 ∙ 3-2 = 35+(-2) = 35-2 = 33 = 27
This happens because the exponent is the indicator of how many times the base number must be multiplied by itself. Therefore, the final exponent will be the addition or subtraction of the exponents that have the same base.
4. Division of powers with the same base or quotient of two powers with the same base
The quotient of two powers with the same base is equal to raising the base according to the difference of the exponent of the numerator minus the denominator. The base must be different than 0.
Examples:
5. Power of a product or Distributive Law of Potentiation with respect to Multiplication
This law establishes that the power of a product must be raised to the same exponent (n) in each of the factors.
Examples:
(a ∙ b ∙ c)n = an ∙ bn ∙ cn
(3 ∙ 5)3 = 33 ∙ 53 = (3 ∙ 3 ∙ 3) (5 ∙ 5 ∙ 5) = 27 ∙ 125 = 3375.
(2ab)4 = 24 ∙ a4 ∙ b4 = 16 a4b4
6. Power of another power
It refers to the multiplication of powers that have the same bases, from which a power of another power is obtained.
Examples:
(am)n = am∙n
(32)3 = 32∙3 = 36 = 729
7. Law of the negative exponent
If you have a base with a negative exponent (an), you must take the unit divided by the base that will be raised with the sign of the exponent positive, that is, 1/an. In this case, the base (a) must be different from 0, a ≠ 0.
Example: 2-3 expressed as a fraction becomes:
You may be interested in Laws of exponents.
Laws of radicals
The law of radicals is a mathematical operation that allows us to find the base through the power and the exponent.
Radicals are square roots that are expressed in the following way √, and consist of obtaining a number that, multiplied by itself, results in what is in the numerical expression.
For example, the square root of 16 is expressed as follows: √16 = 4; This means that 4.4 = 16. In this case it is not necessary to indicate the exponent two in the root. However, in the rest of the roots yes.
For example:
The cube root of 8 is expressed as follows: 3√8 = 2, that is, 2 ∙ 2 ∙ 2 = 8
Other examples:
n√1 = 1, since every number multiplied by 1 is equal to itself.
n√0 = 0, since every number multiplied by 0 is equal to 0.
1. Radical cancellation law
A root (n) raised to the power (n) cancels.
Examples:
(n√a )n = a.
(√4 )2 = 4
(3√5 )3 = 5
2. Root of a multiplication or product
A root of a multiplication can be separated as a multiplication of roots, regardless of the type of root.
Examples:
3. Root of a division or quotient
The root of a fraction is equal to the division of the root of the numerator and the root of the denominator.
Examples:
4. Root of a root
When there is a root within a root, the indices of both roots can be multiplied in order to reduce the numerical operation to a single root, and the radical is maintained.
Examples:
5. Root of a power
When there is a number high in an exponent within a root, it is expressed as the number raised to the division of the exponent by the index of the radical.
Examples:
See also Algebra.