Simplifying Radical Expressions

1
Simplifying Radical Expressions

• For a radical expression to be simplified it has
  to satisfy the following conditions
• The radicand has no factor raised to a power
  greater than or equal to the index. (EXThere are
  no perfect-square factors.)
• The radicand has no fractions.
• No denominator contains a radical.
• Exponents in the radicand and the index of the
  radical have no common factor, other than one.

2
Converting roots into fractional exponents

• Any radical expression may be transformed into an
  expression with a fractional exponent. The key is
  to remember that the fractional exponent must be
  in the form

For example
3
Negative Exponents

• Remember that a negative in the exponent does not
  make the number negative!
• If a base has a negative exponent, that indicates
  it is in the wrong position in fraction. That
  base can be moved across the fraction bar and
  given a postive exponent.

EXAMPLES
4
Simplifying Radicals by using the Product Rule

• If are real numbers and m is a
  natural number, then

Examples This one can not be
simplified any further due to their indexes (2
and 3) being different!
So, the product of two radicals is the radical of
their product!
5
Simplifying Radicals involving Variables

• Examples
• This is really what is taking place, however, we
  usually dont show all of these steps! The
  easiest thing to do is to divide the exponents of
  the radicand by the index. Any whole parts come
  outside the radical. Remainder parts stay
  underneath the radical.
• For instance, 3 goes into 7 two whole times..
  Thus will be brought outside the
  radical. There would be one factor of y remaining
  that stays under the radical.

Lets get some more practice!
6
Practice
EX 1
The index is 2. Square root of 25 is 5. Two goes
into 7 three whole times, so a p3 is brought
OUTSIDE the radical.The remaining p1 is left
underneath the radical.
EX 2
The index is 4. Four goes into 5 one whole
time, so a 2 and a are brought OUTSIDE the
radical. The remaining 2 and a are left
underneath the radical. Four goes into 7 one
whole time, so b is brought outside the radical
and the remaining b3 is left underneath the
radical.
7
Simplifying Radicals by Using Smaller Indexes

• Sometimes we can rewrite the expression with a
  rational exponent and reduce or simplify using
  smaller numbers. Then rewrite using radicals with
  smaller indexes

More examples EX 1 EX 2
8
Multiplying Radicals with Difference Indexes

• Sometimes radicals can be MADE to have the same
  index by rewriting first as rational exponents
  and getting a common denominator. Then, these
  rational exponents may be rewritten as radicals
  with the same index in order to be multiplied.

9
Applications of Radicals

• There are many applications of radicals. However,
  one of the most widely used applications is the
  use of the Pythagorean Formula.
• You will also be using the Quadratic Formula
  later in this course!
• Both of these formulas have radicals in them. To
  learn more about them you may go to
• Pythagorean TheoremWhat is the
  Pythagorean Formula?Quadratic Formula