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Project 4: Simplifying Expressions with Radicals
Defintion
A radical, which we symbolize with the sign √ , is just the opposite of an exponent. In this class, the only radical we will focus on is square roots, which are the opposite of squaring, or raising something to the second power. The square root is just defined as the number (or expression) that would have to be squared in order to result in the number (or expression) under the radical sign. In math, this same idea can be expressed this way:
ܽ ଶ^ ↔ ܾൌ ܾ√ ܽൌ This means, for example, that the following must be true:
ܽඥ ଶ^ ܽൌ Remember that ܽ and ܾ here are standing in for ANY number or expression. So, for example:
ݔ√ ଶ^ ݔ ൌ ඥሺ ݍെ 1ሻ^ ଶ^ ൌ ݍെ 1 ܽඥሺ^ ିܾସ^ ଷ^ ሻ^ ଶ^ ܽൌ ିܾସ^ ଷ
Now you try! Simplify the following expressions:
ݖ√ ଶ^ ൌ
ඥሺ5 ݔെ ݕሻ ଶ^ ൌ
ඥሺ2݉ି ି݊ଷ^ ଵ^ ሻ ଶ^ ൌ
ݔඥሺ2 ଶ^ ݕ െ ݕݔ3 ଶ^ ሻ ଶ^ ൌ
What is “legal” when simplifying radicals? Because radicals are just a different way of writing exponents, the same kinds of rules apply about what we can and can’t do when simplifying radicals. Let’s see how each rule for exponent expressions applies to radicals (below we assume that ്ܾ 0 ): Exponent expression Radical expression ሻ࢈ࢇሺ ^ ࢇ ൌ ^ ࢈ This works because ܾܽሺ^ ሻ^ ଶ^ ܾܽሺ ൌ^ ܾܽሻሺ^ ܽൌ ܾ⋅ ܽ⋅ ܾ⋅ ܽൌ ሻ ⋅ ܽ⋅ ܾ⋅ ܾൌ ሺ ܽ⋅ ܽሻ ⋅ ሺ ܾ⋅ ܾሻ ൌܽ ܾଶ^ ଶ.
ൌ
This works because ቀ ቁ
ଶ ൌ ቀ ቁ ቀ ቁ ൌ ⋅⋅ ൌ ^
మ మ
ሺ ࢇ ࢈ሻ ്^ ࢇ ^ ࢈ ^!
This is true because ܾ ܽሺ ሻ^ ଶ^ ൌ ሺ ܾܽ ܾ ܽሻሺ ܽൌ ሻ ଶ^ ܾܽ ܾ ܾܽ ଶ^ ܽൌ ଶ^ ܾ ࢈ࢇ ଶ.
Radicals with different bases CANNOT be added, and a radical CANNOT be distributed over ADDITION! ࢇ ^ ࢇ ^ ࢇሻ ሺ ൌ This is true because we are simply combining like terms. For example, ܽ2 ଶ^ ܽ3 ଶ^ ܽ5 ൌ ଶ
Note: When these are added, the radical part stays the same , because this is always what happens with like terms—we are simply counting up how many of the ܽ√ ’s there are. ࢇ ^ ࢈ ്^ ࢇሻ ሺ ^ ࢈ These terms cannot be combined because they are NOT like terms —the exponent part of the term is NOT the same. For example, ܽ2 ଶ^ ܾ3 ଶ്^ ܽ5 ܾଶ^ ଶ^!
്࢈√ ࢇ√ ࢈ࢇሻ√ ሺ or ࢈ ࢇሻ√ ሺ Radicals with different bases CANNOT be added together because they are NOT like terms!
So, for example, the following ways of rewriting radical expressions are correct :
√଼√ସ ൌ ට଼ସ
ݔඥ ଶ^ ݔ√ ൌ ݕ ଶ^ ݕඥ 4√3 െ 2√3 ൌ 2√ ට ௬^
ర ௬ మ^ ൌ^
ඥ௬ ర ඥ௬ మ (^) √2 √2 ൌ √2 ⋅ 2 And the following rewriting attempts are ALL incorrect!
ݔඥ ଶ^ 2 ݕൌ √ݔ ଶ^ ඥ2ݕ (^) √1 √3 ൌ √1 3 4√3 2√3 ൌ 6√ 4√3 2√3 ൌ 6√ 2√2 4ඥሺ3ሻ ൌ 6√2 ⋅ 3 2√2 4ඥሺ3ሻ ൌ 6√2 3
Now you try! Use the allowed rules to rewrite each of the following expressions. If you cannot rewrite the expression using one of the rules above, write CANNOT REWRITE as your answer:
ݔඥ ଶ^ ݕ ଶ^ ൌ
Rewriting radical expressions Now that we understand the basic definitions and rules for what we can do with radicals, we want to apply these rules to rewriting radical expressions, usually with the goal of simplifying. The basic rule for simplifying radicals is this:
If we can rewrite what is UNDER the square root sign as a list of SQUARES that are being either MULTIPLIED or DIVIDED (NOT added or subtracted ), then we can simplify the parts under the radical that are written as squares.
For example:
(^) √28 ൌ √2 ⋅ 2 ⋅ 7 ൌ ඥሺ2 ⋅ 2ሻ ⋅ 7 ൌ √2ଶ^ ⋅ 7 ൌ √2ଶ^ ⋅ √7 ൌ 2√ 2√80 ൌ 2√2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5 ൌ 2ඥሺ2 ⋅ 2ሻ ⋅ ሺ2 ⋅ 2ሻ ⋅ 5 ൌ 2√2ଶ^ ⋅ 2ଶ^ ⋅ 5 ൌ 2√2ଶ^ ⋅ √2ଶ^ ⋅ √5 ൌ 2 ⋅ 2 ⋅ 2 ⋅ √5 ൌ 8√ (^) √99 ൌ √3 ⋅ 3 ⋅ 11 ൌ ඥሺ3 ⋅ 3ሻ ⋅ 11 ൌ √3ଶ^ ⋅ 11 ൌ 3√ (^) √28 √63 ൌ √2 ⋅ 2 ⋅ 7 √3 ⋅ 3 ⋅ 7 ൌ ඥሺ2 ⋅ 2ሻ ⋅ 7 ඥሺ3 ⋅ 3ሻ ⋅ 7 ൌ √2ଶ^ ⋅ 7 √3ଶ^ ⋅ 7 ൌ √2ଶ^ √7 √3ଶ^ √ ൌ 2√7 3√ ൌ 5√ 5√75 െ √12 2√27 ൌ 5√3 ⋅ 5 ⋅ 5 െ 1√2 ⋅ 2 ⋅ 3 2√3 ⋅ 3 ⋅ 3 ൌ 5ඥ3 ⋅ ሺ5 ⋅ 5ሻ െ 1ඥሺ2 ⋅ 2ሻ ⋅ 3 2ඥሺ3 ⋅ 3ሻ ⋅ 3 ൌ 5√3 ⋅ 5ଶ^ െ 1√2ଶ^ ⋅ 3 2√3ଶ^ ⋅ 3 ൌ 5√3√5ଶ^ െ 1√2ଶ^ √3 2√3ଶ^ √ ൌ 5 ⋅ √3 ⋅ 5 െ 1 ⋅ 2 ⋅ √3 2 ⋅ 3 ⋅ √ ൌ 25√3 െ 2√3 6√ ൌ 25√3 6√3 െ2√ ൌ 31√3 െ 2√ ൌ 29√ െ3 5√3 െ 5 2√12 ൌ െ3 േ 5 5√3 2√ ൌ െ3 െ5 5√3 2√2 ⋅ 2 ⋅ 3 ൌ െ8 5√3 2ඥሺ2 ⋅ 2ሻ ⋅ 3 ൌ െ8 5√3 2√2ଶ^ ⋅ 3 ൌ െ8 5√3 2√2ଶ^ √ ൌ െ8 5√3 2 ⋅ 2√ ൌ െ8 5√3 4√ ൌ െ8 9√
√ଶ√ √ଵ^
ଶ⋅
ଵ ൌ ට^
ଵଶ
ൌ √5ଶ^ √3 ⋅ 5ଶ
ൌ √5ଶ^ √3√5ଶ
ൌ 3√2ଶ^ െ1√2 െ
Now you try! Simplify the following radical expressions, or if they cannot be simplified, write CANNOT BE SIMPLIFIED: (If you need more space, feel free to do all the work on your own paper!)
(^) √
√ଷହ√ଵ √
