Examples of expression in the following topics:
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- 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:
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- 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.
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- 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:
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- 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:
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- 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:
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- 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.
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- 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.
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- 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:
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- 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.
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- 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: