The length of the two legs of a right triangle are
Guidance
Before we can solve a quadratic equation using square roots, we need to review how to simplify, add, subtract, and multiply them. Recall that the square root is a number that, when multiplied by itself, produces another number. 4 is the square root of 16, for example. 4 is also the square root of 16 because
Example A
Find
a) A calculator.
b) By simplifying the square root.
Solution:
a) To plug the square root into your graphing calculator, typically there is a
b) To simplify the square root, the square numbers must be “pulled out.” Look for factors of 50 that are square numbers: 4, 9, 16, 25... 25 is a factor of 50, so break the factors apart.
Radical Rules
1.
2.
3.
Example B
Simplify
Solution: At first glance, it does not look like we can simplify this. But, we can simplify each radical by pulling out the perfect squares.
Rewriting our expression, we have:
Example C
Simplify
Solution: Multiply across.
Now, simplify the radical.
Intro Problem Revisit We must use the Pythagorean Theorem, which states that the square of one leg of a right triangle plus the square of the other leg equals the square of the hypotenuse.
So we are looking for c such that
Simplifying, we get
Therefore,
Therefore,
Rationalize the Denominator
Often when we work with radicals, we end up with a radical expression in the denominator of a fraction. We can simplify such expressions even further by eliminating the radical expression from the denominator of the expression. This process is called rationalizing the denominator.
Case 1 There is a single radical expression in the denominator 23√ .
In this case, we multiply the numerator and denominator by a radical expression that makes the expression inside the radical into a perfect power. In the example above, we multiply by the
In this case, we multiply the numerator and denominator by a radical expression that makes the expression inside the radical into a perfect power. In the example above, we multiply by the
Guided Practice
Simplify the following radicals.
1.
2.
3.
Answers
1. Pull out all the square numbers.
Alternate Method: Write out the prime factorization of 150.
Now, pull out any number that has a pair. Write it once in front of the radical and multiply together what is left over under the radical.
2. Simplify
Rewrite the expression:
3. This problem can be done two different ways.
First Method: Multiply radicals, then simplify the answer.
Second Method: Simplify radicals, then multiply.
Depending on the complexity of the problem, either method will work. Pick whichever method you prefer.
Vocabulary
 Square Root
 A number, that when multiplied by itself, produces another number.
 Perfect Square
 A number that has an integer for a square root.
 Radical

The
√ , or square root, sign.
 Radicand
 The number under the radical.
Practice
Find the square root of each number by using the calculator. Round your answer to the nearest hundredth.
 56
 12
 92
Simplify the following radicals. If it cannot be simplified further, write cannot be simplified.

18−−√ 
75−−√ 
605−−−√ 
48−−√ 
50−−√⋅2√ 
43√⋅21−−√ 
6√⋅20−−√ 
(45√)2  \begin{align*}\sqrt{24} \cdot \sqrt{27}\end{align*}
 \begin{align*}\sqrt{16} + 2\sqrt{8}\end{align*}
 \begin{align*}\sqrt{28} + \sqrt{7}\end{align*}
 \begin{align*}8 \sqrt{3}  \sqrt{12}\end{align*}
 \begin{align*}\sqrt{72}  \sqrt{50}\end{align*}
 \begin{align*}\sqrt{6} +7 \sqrt{6}  \sqrt{54}\end{align*}
 \begin{align*}8 \sqrt{10}  \sqrt{90}+7\sqrt{5}\end{align*}