Radical Expressions and Rational Exponents
Learning Outcomes
- Convert between radical and exponent notations
Radical Form |
Exponent Form |
Principal Root |
---|---|---|
Example
Fill in the missing cells in the table.Exponent Form | Root Form | Root of a Square | Simplified |
---|---|---|---|
Answer:
Exponent Form | Root Form | Root of a Square | Simplified |
---|---|---|---|
Radical Form |
Exponent Form |
Principal Root |
---|---|---|
Radical Form |
Exponent Form |
---|---|
… | … |
Example
Write as an expression with a rational exponent.Answer: The radical form can be rewritten as the exponent . Remove the radical and place the exponent next to the base.
Writing Fractional Exponents
Any radical in the form can be written using a fractional exponent in the form .Write an Expression with a Rational Exponent as a Radical
In the following examples, we will show how to convert expressions with rational exponents to expressions with a radical.Example
Express in radical form.Answer: Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.
The parentheses in indicate that the exponent refers to everything within the parentheses.
Example
Express in radical form.Answer: Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.
The exponent refers only to the part of the expression immediately to the left of the exponent, in this case x, but not the .
Write an Expression with a Radical as a Rational Exponent
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Example
Express with rational exponents.Answer: Rewrite the radical using a rational exponent. The root determines the fraction. In this case, the index of the radical is , so the rational exponent will be .
Since is outside the radical, it is not included in the grouping symbol and the exponent does not refer to it.
Rational Exponents Whose Numerator is Not Equal to One
Notice that in the previous two examples, the radicands had exponents. We simplified these expressions using factorsing, but we can still convert these radical expressions to expressions with rational exponents. Also, note that all of the numerators for the fractional exponents in the previous examples above were . You can use fractional exponents that have numerators other than to express roots, as shown below.
Radical |
Exponent |
---|---|
… | … |
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Writing Rational Exponents
Any radical in the form can be written using a fractional exponent in the form .Example
Rewrite the radicals using a rational exponent, then simplify your result.Answer: 1. can be rewritten as , so in this case , therefore Simplify the exponent. 2. can be rewritten as , so in this case , therefore
Simplify the expression using rules for exponents.
Try It
[ohm_question]3537[/ohm_question]Example
Rewrite the expressions using a radical.Answer:
- , the numerator is and the denominator is , therefore we will have the third root of x squared,
- , the numerator is and the denominator is , so we will have the seventh root of raised to the fourth power.