27 November 2023

No matter what level you are at learning algebra topics, exponents are part of math exercises. Therefore, it is vital to know what they are and whatHow important is it in a mathematical equation?. These usually appear from the perfect square binomials and trinomials, to indicate the power to which a high term is found.

Therefore, today you will have the opportunity to understand what its definition is, as well as the uses that can be given to exponents within a mathematical or algebraic formula. At the end, you will know the most important aspects and it will be easier for you to solve exercises with less difficulty.

Definition and concept of an exponent in mathematics

It is a mathematical operation that derives from multiplication. So we can say that it is a slightly more complex mathematical form, but that, at the same time, it helps us to simplify an algebraic expression. In a simple way, it is a number which tells us the number of times that we are going to multiply a term by itself. It can be expressed as a whole number, positive or negative, in addition to the fact that fractions can be applied to identify a power or exponent.

In general, the exponents are usually expressed by the number two and three. In such a case, it is said that the term is raised to the square or to the cube respectively. However, this does not mean that a term or elements cannot be raised to another power greater or less than these. The detail of this is that it is not always the case and for practical purposes, it is not very useful in the workplace. Of course, it is very useful to identify which are the powers that have the similar terms, in order to be able toapply the associative property and solve a problem more accurately.

So, in summary, we can define the concept of algebraic exponent as follows:

  • It is a number or index that is located at the top right of a number or term

  • It can be applied to both real numbers and terms with independent variables.

  • Indicates the number of times that term will multiply itself

  • You can provide us with information regarding the degree of a mathematical equation.

  • A term can be raised to a positive, negative, or fractional number.

What is an exponent used for in algebra?

The exponent or the powers, in general, are usually used when in a multiplication with similar terms, it is too large or repetitive. For example, if we have the following multiplication: 20x*(-x)*4(xy). It is much more practical to accommodate this algebraic expression, in such a way that it is as follows: -40yx³. But why do we do it this way?

It is thanks to the propertytocommutative d in multiplication, that I can put the variable y and Y first, followed by the cubed X. Of course, always respecting the law of signs, which tells us that more for less is less. To make it easier to read, I put the negative sign at the beginning of everything, followed by the number, then the variable with the lowest exponent followed by the one with the highest power.

On the other hand, the exponents have a particular use, which is to indicate the degree of a term, a polynomial or an equation. For example, the quadratic equation is so called because its degree is of index two. However, there are equations and polynomials that have a degree more than 2. So you will find mathematical equations of degree three, four, five and even more. The advantage and benefits that algebra gives us, is that the quadratic equation, like many others in mathematics, can be solved. That is, they can be written in a certain way, which makes their use more practical.

What are the characteristics of an exponent in mathematics?

There are two fundamental characteristics that define exponents in mathematics. The first of these is that there must be a basis for a power to exist. That is, a variable number or a term that combines both elements. Example of this can be: 8, X or 9X. Here we have three bases with different elements, so the first condition or characteristics is met. Now, there is the second, which is the power, index or exponent.

The latter indicates both the degree of the term and the number of times it will be repeated. For example, if we have: They are two different cases, but at the same time similar. That is to say, that, for the first case, only the independent variable Y is cubed, while, in the second case, everything that is inside the parentheses is being affected by the power. When there is a base, we can apply one of the different leyes of exponents in algebra. In case you do not know what these laws are, we will detail each one shortly. So you can make a custom form after finishing the article.

What are the laws of exponents in algebra?

There are many laws of which, you have to learn each and every one of them, since you will use it in almost everything that has to do with mathematics. Let’s start from the simplest and we will increase the difficulty.

  1. Any term that is raised to the zero power will be equal to 1.

  2. That number that is raised to the power one, will be equivalent to the same term.

  3. Any number or variable with a negative exponent will be the inverse of the term. That is, the base will divide. For example: X^-1 = 1/X. Salways keeping the base with its exponent.

  4. When there is a multiplication of terms with the same base, their exponents are added and the same base is placed.

  5. When it comes to a division of terms with the same base, the exponents are subtracted and the same base is placed.

  6. If a term with a certain exponent is raised to a power, we proceed to multiply the exponents and leave the same base.

  7. Everything that is inside a parenthesis and is raised to a power X, all the elements will have the same power. In other words, the power outside the parenthesis is distributed among all the internal elements.

  8. If a term is raised to a rational power or fraction, this has its equivalent in a root. The denominator becomes the index of the root, while the numerator is the power of the term or radicand.

The good thing about these properties is that you will find many simple examples where you can better visualize each application according to the corresponding property. Keep in mind that it is very important that you master this perfectly, since it will serve you for almost everything and for any subject that is related to calculus.

What are the parts of exponents in math and algebra?

Unlike other properties in mathematics, this one in particular has only two parts. The first one is the base, which can be expressed with a number, a variable, or a combination of both. The second part is the exponent or also known as power. may be represented by through a number a variable or the combination of both. In addition, the element that is considered as power can be both positive and negative.

Examples of exponents in algebraic terms

The idea that you can learn to deal with exponents is by applying the standard formulas that exist for each case according to the property. For example, any number raised to 0 will give you one as a result. That is, if you have X^0 = 1; 1000⁰ = 1 ; (2X)^0 = 1. If you look closely, there are tons of ways to write the number, as long as you know how to apply the correct property.

Now, to complicate it a bit, imagine that you have the following algebraic expressions: (2X)^-4. According to one of the power properties, this is equivalent to writing: 1/ (2X)^4. This is due to one of the properties, which tells us that everything term raised to a negative power, can be written as a fraction, keeping the same power, but in a positive way. To understand this better, look at it as if a negative exponent is the inverse. By having such a term, in the numerator, its inverse would be a fraction. Now, if the exponent is in the denominator, its inverse would be a numerator.

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