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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. (EX:There are no perfect-square factors.) The radicand has no fractions.

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simplifying radical expressions

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. (EX:There 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 nocommon factor, other than one.

converting roots into fractional exponents
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

=

negative exponents
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:

simplifying radicals by using the product rule
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!

simplifying radicals involving variables
Simplifying Radicals involving Variables:
  • Examples:

This is really what is taking place, however, we usually don’t 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.

Let’s get some more practice!

practice
Practice:

EX 1:

The index is 2. Square root of 25 is 5. Two goes into 7 three “whole” times, so a

p3is 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.

simplifying radicals by using smaller indexes
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:

multiplying radicals with difference indexes
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.
applications of radicals
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