CHAPTER TEN – RADICAL EXPRESSIONS
10.1 Introduction to Radical Expressions
The ______ of a positive number x, written , is a number whose square is x.
Examples: since
since
since
In the the x is called the ______and the is called the ______.
Note that every positive number has two square roots – one ______and one ______.
When we use the symbol, we are always referring to the positive (also called the ______) root.
(If we want to indicate the negative root, we must place a negative symbol in front of the radical.)
Example: 25 has two square roots, +5 and –5:
The square of any integer is called a ______.
Make a list here of the perfect squares for integers 1 through 20.
Any perfect square can be factored into a product of prime factors raised to an even power:
Examples:
If a number is NOT a perfect square, then the square root of the number is a non-repeating decimal. We call this an ______.
Example:
The pattern would never repeat, and with a stronger computer we could find more decimal places.
TO SIMPLIFY RADICAL EXPRESSIONS:
Use the following property:
Our goal in simplifying radical expressions is to evaluate any factors of the radicand which are perfect squares and leave the factors which are not perfect squares alone under the radical.
STEP 1: Break off all factors of the radicand that are perfect squares. This means any:
· integer perfect squares as defined previously
· variable factors with even exponents
STEP 2: Write each perfect square factor under its own square root sign. Leave all of the remaining "non-perfect-square" factors together in their own square root.
STEP 3: Evaluate the square root for each of the perfect squares.
We'll start out with some examples involving constants only:
Example:
How did we decide on factoring 108 into 36 and 3 instead of, for example, 4 and 27 or maybe 9 and 12? Because we want the biggest perfect square factor we can find. When you start trying to simplify a square root, say to yourself: “What is the largest perfect square factor of 108?” and then factor it out and take the square root of that piece.
Example:
“What is the largest factor of 45 which is a perfect square?” ______
Factor it out and evaluate the square root on that piece only:
Examples:
APPLYING THE RULE TO MONOMIALS CONTAINING VARIABLE FACTORS
If the exponent on a variable is EVEN, it is a perfect square.
Examples:
To evaluate the square root of variables with even exponents, simply divide the exponent by 2!!
Example:
Examples:
If the exponent on a variable is ODD, we treat it with the same rule we used for numerical radicands. “Find the largest perfect square factor and factor it out”. Except now the largest perfect square factor is the exponent one lower.
Ex.: Since has an even exponent, it’s a perfect square.
Since has an even exponent, it’s also a perfect square
Examples:
In each case, remember, we want the BIGGEST perfect square factor:
· If the power is even, its already a perfect square
· If the power is odd, break off a single x (or y or p) and drop the exponent by one.
Now, we can evaluate square roots of monomials as follows:
If you are comfortable doing some steps together, we have the following:
Keep this in mind when trying to remember how to simplify:
· The “Perfect” factors (perfect squares) get to go out as a group and have fun
· The “Oddball” factors have to stay home – so they get together and have a party at one house
Examples:
10.2 Addition and Subtraction of Radicals
The rule for addition and subtraction of radicals is almost identical to the rule for adding and subtracting variable expressions: Combine Like Terms à Combine Common Radicals
This is done by adding the coefficients and keeping the radical expression:
Example: just as 3x + 7x combines to give 10x,
If the radicals do not have the same radicand in simplest form we cannot add them.
Example: does not simplify because the radicands, 5 and 7, do not match
Example: looks as though it should not simplify, but the two radicals are not in simplest form, so simplify them and then add
Examples:
10.3 Multiplication and Division of Radical Expressions
The rule for multiplication we have already seen, but used it in the reverse of how we will use it here:
The rule for multiplying radicals with lead coefficients is simply stated:
Multiply the outsides, multiply the insides
Example: Multiply and simplify
ALWAYS MULTIPLY THE RADICALS FIRST, THEN SIMPLIFY THE RESULT
Examples:
When the multiplication problem involves a “polynomial”-like radical, we use the same rules for multiplying that we used for polynomials:
Distribute:
Examples:
FOIL:
Note: The two middle terms do not have common radicals after being simplified, so they do not combine.
Examples:
CONJUGATES: (a + b)( a – b)
Foil is performed the same way as above even when we have conjugates, but we will notice that the two middle terms drop out of the product.
Examples:
DIVISION OF RADICALS
Note that the above property can be used either forwards or backwards:
Examples:
RATIONALIZING DENOMINATORS:
When a radical appears in the denominator of a fraction, we are asked to rationalize it by multiplying top and bottom by the same radical. This will create an INTEGER or monomial in the denominator.
Example: Rationalize (or simplify)
Multiply top & bottom by ®
Examples: