1. Radical Notation and Rational Exponents
Section R.7
Radical Notation and Rational Exponents

2. R.7 Radical Notation and Rational Exponents
Simplify radical expressions.
Rationalize denominators or numerators in rational expressions.
Convert between exponential and radical notation.
Simplify expressions with rational exponents.

3. Notation
A number c is said to be a square root of a if c2 = a. Similarly, c is a cube root of a if c3 = a.
nth Root
A number c is said to be an nth root of a if cn = a.
The symbol denotes the nth root of a. The symbol is called a radical. The number n is called the index.
Any positive number has two square roots, one positive and one negative. For any even index, a positive number has two real-number roots. The positive root is called the principal root.

4. Examples Simplify each of the following: a) , because 62 = 36.
b) because 62 = 36 and
c) , because (−2)3 = −8. d)
e) Is not a real number, because we cannot find a real number that can be raised to the fourth power to get −16.

5. Properties of Radicals
Let a and b be any real numbers or expressions for which the given roots exist. For any natural numbers m and n (n  1):
1. If n is even,
2. If n is odd, 3. 4. 5.

6. Examples a) b) c) d) e)

7. Examples
f) h) g)

8. The Pythagorean Theorem
The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse:
a2 + b2 = c2.

9. Example
Building a Doll House. Roy is building a doll house for his granddaughter. The doll house measures 26 in. across, and the slanted side of the roof measures 15 in. Find the height of the roof.

10. Example
Solution
The height of the room is about 7.5 in.

11. Rationalizing Denominators or Numerators
There are times when we need to remove the radicals in a denominator or a numerator. This is called rationalizing the denominator or rationalizing the numerator. It is done done by multiplying by 1 in such a way as to obtain a perfect nth power.
Example
Rationalize the denominator.

12. Rationalizing Denominators or Numerators
Pairs of expressions of the form
are called conjugates. The product of a pair contains no radicals and can be used to rationalize a denominator or numerator.

13. Rationalizing Denominators or Numerators
Example
Rationalize the numerator.

14. Rational Exponents
For any real number a and any natural numbers m and n, for which exists,

15. Example Convert to radical notation and, if possible, simplify. a) b)

16. Example
Simplify and then, if appropriate, write radical notation for each of the following. a) b) c)