## Basic Terms of Algebra

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**BASIC TERMS OF ALGEBRA**

In this lesson, we will learn the basic terms used in Algebra. The good news is that you already know many of these basic terms from learning Arithmetic.

The basic terms most used in Algebra are constants, variables, operators, coefficients, powers, expressions and equations.

We already discussed the first two terms, constants and variables, in the previous lesson. Remember that a constant is a value that never changes like 5, 9, 8.9, or ⅓. In other words, constants are simply numbers.

The value of a variable is constantly changing, and variables are represented by letters such as x, y, z or any other symbol.

The next term we will learn is operator. Operators are used to define the relationship between constants and variables. There are a number of operators used in Algebra but here we will learn some of the basic operators of Algebra. Every operator is represented by a unique symbol.

Let’s discuss the operators one by one.

The first one is addition. If we want to add variables x and y, we write x + y and read it as “x plus y.”

The second operator is subtraction. Suppose we want to find the difference of two variables: x and y, we write x – y and read it as “x minus y.”

The third operator is multiplication. If we want to find the product of two variables x and y, we write (x)(y) or x * y and read it as “x times y.”

The fourth operator is division. If we want to divide variables x by y, we write x / y and we read it as “x divided by y.”

Other operators like >, <, ≤, ≥ read as greater than, less than, greater than or equal, and less than or equal. These operators are used to compare two variables.

Now, let’s go back to the basic terms of Algebra. The next term is coefficient. A coefficient is a number or a constant value multiplied by a variable.

For example, in the term 5x, 5 is the coefficient and x is the variable.

The next term is exponent, which is also called power. The power of a number defines how many times a number is repeated as a factor. Take for example,

x*x*x*x*x.

Here we have five factors all equal to x. We can rewrite it in exponent form as x^{5} and read it as “x raised to the power of five.” The factor being repeated is called base and the number of times it is repeated as a factor is called the exponent or power. In this example, x is the base and 5 is the power.

Some exponents have their own labels. For example, the power 2 is usually read as “squared.” So x^{2} can be read as “x squared” or “x raised to the power of two.” Another example is the power 3. x^{3} can be read as “x cubed” or “x raised to the power of three.”

There are some rules that need to be followed when dealing with numbers containing powers.

The first one states that any number raised to the power 0 is always equal to 1, i.e., n^{0} = 1. Some examples are 3^{0}=1, 190^{0 }=1 and 1850^{0}=1.

The second rule is if any exponential terms with the ,same base are multiplied the base remains the same and their powers are added.

For example,

x^{4} * x^{3} * x^{2}.

Here we have three exponential terms all having the same base which is x. Then we add the powers and write

x^{4+3+2}

which leads to the simplified form

x^{9}.

Next rule is if any exponential terms with the same base are divided, the base remains the same and the power is the difference between the powers of the exponential terms, i.e., the power of the numerator minus the power of the denominator. To illustrate this, let’s consider the expression

z^{5}/z^{2}.

Both terms have the same base which is 3. So we keep 3 as the base. Then we get the difference of the powers of the numerator and denominator. Now we can rewrite the expression as

z^{5-2}

which results to

z^{3}.

Before going to the next rule, let’s learn how to get the reciprocal of a number. In this process we simply write the number upside down, e.i., write the numerator as denominator and the denominator as numerator. Take for example finding the reciprocal of 5. By convention, the denominator of a whole number is one, so 5 can be written as

5/1.

We write this upside down, put 5 in the denominator and 1 in the numerator, to get

1/5.

The reciprocal of 5 is 1/5.

Now we go to the next rule involving exponential terms. This rule tells that any exponential term with a negative exponent is equal to the reciprocal of the same term with the negative sign in the power omitted.

For example,

x^{-5}.

The reciprocal of this exponential term is

1/x^{-5}.

Then we drop the negative sign in the exponent which makes it

1/x^{5}.

So what we have is the relation

x^{-5} = 1/x^{5}.

The next rule is for an exponential term raised to a power. In this case, the exponents are multiplied. For example,

(y^{2})^{4}.

Here the exponential term y^{2} is raised to the power of 4. What we do is multiply both the powers and write

y^{2}*^{4}.

This leads to the simplified form

y^{8}.

Now that we’ve gone over some of the rules for working for exponents, we’ll learn what expressions are.

An expression is a single mathematical term or combination of two or more terms separated with any operator +, -, * or /.

For example,

10

5x

3a + 6b

(10xy – 2y) / y

are all expressions.

In solving or simplifying mathematical expressions with multiple operators, we need to identify which operator to work on first. We call this the “order of operations”.

First of all, if there are any brackets in the expression, the problem inside the brackets needs to be solved first. Second in the order of operations is working on the powers or “exponents”. The third step is to complete any division or multiplication. It doesn’t matter whether division or multiplication is done first. And finally the last step is to complete any addition or subtraction.

We need to follow this order of operations to get the correct answer of an expression.

Let’s take a look at an example together and practice our order of operations.

We solve for the expression

2 + 5^{2} * ( 9 / 3 – 1).

First in the order of operations is to work on expressions inside parentheses. So we solve first

9/3 -1.

Here we have two operations, division and subtraction. Remember that in the order, division comes first. So we have

3 -1

which gives us

2.

The expression

2 + 5^{2} * ( 9 / 3 – 1)

can then be written as

2 + 5^{2} * 2.

Now that we’re done with the terms inside the parentheses, we work on the exponential term 5^{2}.

Remember that 5^{2} means 5 times 5 which is equal to 25. So the expression becomes

2 + 25*2

Here we are left with two operators, addition and multiplication. Following the order of operations, the multiplication should be done first which results to

2 + 50.

Finally, adding the two terms left, the answer is

52.

The last topic of the lesson we are going to discuss is about equations. Equations are two expressions separated by an equal sign.

Every equation has a left hand side and a right hand side. As both sides are separated by an “equal to” symbol which is written as =, they must eventually satisfy the equality criteria, that is, they must work out to the same answer.

For example,

2x + 3y = 15.

The left hand side expression is equal to 15. We will discuss more about expressions and equations in the next lesson. It will be very interesting as we will learn how to write expressions and equations from phrases or sentences.

In summary, we have discussed the basic terms in algebra which include constants, variables, operators, coefficients, powers, expressions and equations. There are four basic operators namely addition, subtraction, multiplication and division. We also discussed some rules in combining exponential terms. Most importantly, we learned that we have to follow a certain order when multiple operators are present in an expression. This is what we call the “order of operations” which can easily be recalled by the acronym PEMDAS which stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. Remember that multiplication and division fall in the same order which means it is acceptable work on divisions first followed by the multiplications. In similar manner, addition and subtraction fall in the same order.