# Simplifying Radical Expressions

In the last section every number under the square root
symbol was a perfect square. When this

happens the radical sign disappears and entire square root is replaced with with
a rational number.

When you find the square root of a perfect square there will not be a square
root in the answer.

was replaced with a 3 | was replaced with a 5 | was replaced with a 6/7 |

Most of the time the number under the square root is not a
perfect square. The square root of any

number that is not a perfect square can't be replaced by any fraction, decimal
or whole number. This

type of number is an irrational number. It represents a decimal that never ends
or repeats. Since you

cannot write such a decimal you cannot replace such a square root as a number
without a radical sign.

cannot be replaced | cannot be replaced | cannot be replaced |

with a decimal or fraction, | with a decimal or fraction, | with a decimal or fraction, |

it stays | it stays | it stays |

**Some square roots can be reduced to the
square root of a smaller number**

If the number under the square root is not a
perfect square then it cannot replaced with a number

without a radical sign. It may be replaced with an expression that has a smaller
number under the

radical sign.

can be replaced by | can be replaced by | can be replaced by |

We call the process of replacing one square root
expression with another square root expression that

has a smaller number under the radical sign reducing or **simplifying the
square root.**

**Multiplication Rule for Square Roots**

This rule allows you to factor a number under
the square root into two (or more) factors and write the

factors as a product under two separate square roots. If one of the factors is a
perfect square then that

square root can be reduced leaving you with a number outside the radical times a
smaller square root

then the original square root you started with.

**Simplifying Radicals using PERFECT
SQUARE FACTORS**

If you can find the **largest** perfect square factor
of the radicand then reducing the radical expression is

a short process. It requires that you find the largest perfect square that is a
factor of the original

radicand

**Look for the largest perfect square that is a factor of
the radicand. Factor than reduce.**

Example 1 |
Example 2 |
Example 3 |

**Example 1 Note:** To use this technique you must
factorasand
not as

Example 4 |
Example 5 |
Example 6 |

**Example 6 Note:** To use this technique you must
factorasand
not as

The factor must be the largest square root factor

Example 7 |
Example 8 |
Example 9 |

Example 10 |
Example 11 |
Example 12 |

**Simplifying Radicals using PAIRS OF
FACTORS**

If you can find the largest perfect square
factor of the radicand then reducing the radical expression is a

short process. Many students cannot find the largest perfect square factor or
they do not want to take

the extended time this may take. There is a alternate approach that is favored
by many students.

Two of the same factors under a square root
form a perfect square. This means that if you have a pair

of the same factors under a square root they can be reduced to a rational
number.

a pair of 2's under a square root reduce to the whole number 2 |
a pair of 3's under a square root reduce to the whole number 3 |
a pair of 5's under a square root reduce to the whole number 5 |

This fact allows us to use the Multiplication Rule for
Square Roots to completely factor a radicand into

its many factors and then take out the pairs of same factors

Example 1 |
Example 2 |
Example 3 |

completly factor 24 | completly factor 24 | completly factor 24 |

put pairs of the same factor under thier own square root |
put pairs of the same factor under thier own square root |
put pairs of the same factor under thier own square root |

the pair of 2's can reduced | the pair of 2's can reduced | the pair of 2's can reduced |

**Completely factor the radicand and take out all the
pairs of factors:**

Example 4 |
Example 5 |
Example 6 |

Example 7 |
Example 8 |
Example 9 |

Example 10 |
Example 11 |
Example 12 |