expression

Examples of expression in the following topics:

• Simplifying Exponential Expressions

  • Multiplying exponential expressions with the same base: $a^m \cdot a^n = a^{m+n}$
  • Previously, we have applied these rules only to expressions involving integers.
  • The same rule applies to expressions with variables.
  • Now apply the rule for dividing exponential expressions with the same base:
  • To simplify the first part of the expression, apply the rule for multiplying two exponential expressions with the same base:

• Simplifying, Multiplying, and Dividing Rational Expressions

  • Performing these operations on rational expressions often involves factoring polynomial expressions out of the numerator and denominator.
  • As a first example, consider the rational expression $\frac { 3x^3 }{ x }$.
  • We follow the same rules to multiply two rational expressions together.
  • Dividing rational expressions follows the same rules as dividing fractions.
  • The same applies to dividing rational expressions; the first expression is multiplied by the reciprocal of the second.

• Simplifying Radical Expressions

  • A radical expression that contains variables can often be simplified to a more basic expression, much as can expressions involving only integers.
  • Expressions that include roots are known as radical expressions.
  • A radical expression is said to be in simplified form if:
  • For the purposes of simplification, radical expressions containing variables are treated no differently from expressions containing integers.
  • This follows the same logic that we used above, when simplifying the radical expression with integers:

• Adding and Subtracting Algebraic Expressions

  • Every algebraic expression is made up of one or more terms. 
  • Terms in these expressions are separated by the operators $+$ or $-$.
  • For instance, in the expression $x + 5$, there are two terms; in the expression $2x^2$, there is only one term.
  • The same rules apply when an expression involves subtraction.
  • The expression therefore simplifies to:

• Negative Exponents

  • This rule makes it possible to simplify expressions with negative exponents.
  • For example, consider the rule for multiplying two exponential expressions with the same base.
  • Note that the rule for raising an exponential expression to another exponent can be applied:
  • Recall that the rule for multiplying two exponential expressions with the same base can be applied.
  • Therefore, we can simplify the expression inside the parentheses:

• Introduction to Radicals

  • Radical expressions yield roots and are the inverse of exponential expressions.
  • Mathematical expressions with roots are called radical expressions and can be easily recognized because they contain a radical symbol ($\sqrt{}$).
  • Since roots are the inverse operation of exponentiation, they allow us to work backwards from the solution of an exponential expression to the number in the base of the expression.
  • In this expression, the symbol is known as the "radical," and the solution of 7 is called the "root."
  • Finding the value for a particular root can be much more difficult than solving an exponential expression. 

• The Order of Operations

  • The order of operations is an approach to evaluating expressions that involve multiple arithmetic operations.
  • The order of operations is a way of evaluating expressions that involve more than one arithmetic operation.
  • For example, when faced with the expression $4+2\cdot 3$, how do you proceed?
  • In order to be able to communicate using mathematical expressions, we must have an agreed-upon order of operations so that each expression is unambiguous.
  • This expression correctly simplifies to 9.

• Rational Algebraic Expressions

  • When applying this strategy to rational expressions, first look at the denominators of the two rational expressions and see if they are the same.
  • This requires factoring algebraic expressions.
  • For example, consider the expression $2x^2 + 4$.
  • This expression therefore has two factors: $2$ and $(x^2 + 2)$.
  • The rational expressions therefore become:

• Equations and Inequalities

  • An equation states that two expressions are equal, while an inequality relates two different values.
  • An equation is a mathematical statement that asserts the equality of two expressions.
  • This is written by placing the expressions on either side of an equals sign (=), for example:
  • Equations often express relationships between given quantities—the knowns—and quantities yet to be determined—the unknowns.
  • The process of expressing the unknowns in terms of the knowns is called solving the equation.

• Introduction to Exponents

  • For example, the expression $b^3$ represents $b \cdot b \cdot b$.
  • Here, the exponent is 3, and the expression can be read in any of the following ways:
  • Now that we understand the basic idea, let's practice simplifying some exponential expressions.
  • Let's look at an exponential expression with 2 as the base and 3 as the exponent:
  • Let's look at another exponential expression, this time with 3 as the base and 5 as the exponent: