Addition and Subtraction of Radicals
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3.1 Addition and Subtraction of Radicals In the study of polynomials, two terms are called "like" terms if the variables and the exponents of the variables are the same. For example: • the terms 2 x 2 and16 x 2 are "like" terms • the terms 2 x and 16x 3 are not "like" terms The situation with square root radicals is the same. Two square root radicals are called "like" radicals if the radicands are the same. Therefore: • the radicals 2 √ 3and−6 √ 3 are "like" radicals • the radicals √ 6and √ 3 are not "like" radicals Just as with polynomials, "like" radicals can be combined by addition and subtraction. Part of simplifying radical expressions is combining all the "like" terms. Example 1
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Radical Operations
Transcript of Addition and Subtraction of Radicals
3.1Addition and Subtraction of Radicals
In the study of polynomials, two terms are called "like" terms if the variables and the exponents of the variables are the same.
For example:
the terms are "like" terms
the terms are not "like" terms
The situation with square root radicals is the same.
Two square root radicals are called "like" radicals if the radicands are the same.
Therefore:the radicals are "like" radicals
the radicals are not "like" radicals
Just as with polynomials, "like" radicals can be combined by addition and subtraction.
Part of simplifying radical expressions is combining all the "like" terms.
Example 1
Simplify .
Solution:
Write the original expression.
Combine like radicals.=
Simplify.=
Since are not "like" radicals, no further simplification can occur.
The distributive property is also used when simplifying radical expressions. The same rules apply as when working with polynomials or rational expressions.
The Distributive Property
The product of a and is given by:
or .
The product of a and is given by:
or .
Example 2
Simplify the radical expression .
Solution:
Write the original expression.
Apply the distributive property.=
Group the "like" terms.=
Combine "like" terms.=
Radicals such as do not appear to be "like" radicals. In this case, it is necessary to change to a mixed radical.
and now become "like" radicals.
Example 3
Simplify the expression by first changing all of the radicals to the simplest possible mixed radical form.
Solution:
Change each radical to the simplest possible mixed radical form.
is in its simplest form
Write the original expression.
Express in simplest mixed radical form.=
Combine like terms.=
Simplify.=
Example 4
Simplify and then evaluate for .
Solution:
Write the original expression.
Change each radical to a mixed radical.=
Combine the like terms.=
Substitute .=
Simplify.=
Apply the distributive property.=
The and 44 are not "like" and cannot be further combined.
Exercise 3.1
1.Simplify.
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2.Simplify.
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3.Simplify.
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4.Simplify.
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5.Prove that is false by finding just one set of values for x and y for which the equation does not hold.
6.Simplify. Assume that all variables are non negative.
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