Estimate square roots'

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Objectives
Estimate square roots. Simplify, add, subtract,
multiply, and divide square roots.
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Vocabulary
radical symbol radicand principal
root rationalize the denominator like radical
terms
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• The side length of a square is the square root
  of its area. This relationship is shown by a
  radical symbol . The number or expression
  under the radical symbol is called the radicand.
  The radical symbol indicates only the positive
  square root of a number, called the principal
  root. To indicate both the positive and negative
  square roots of a number, use the plus or minus
  sign ().

or 5
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• Numbers such as 25 that have integer square
  roots are called perfect squares. Square roots of
  integers that are not perfect squares are
  irrational numbers. You can estimate the value of
  these square roots by comparing them with
  perfect squares. For example, lies between
  and , so it lies between 2 and 3.

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Example 1 Estimating Square Roots
Estimate to the nearest tenth.
Find the two perfect squares that 27 lies between.
Find the two integers that lies between
.
Try 5.2 5.22 27.04
Too high, try 5.1.
5.12 26.01
Too low
Check On a calculator 5.1961524
5.1 rounded to the nearest tenth.
?
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Check It Out! Example 1
Estimate to the nearest tenth.
Find the two perfect squares that 55 lies
between.
Find the two integers that lies between
.
Try 7.2 7.22 51.84
Too low, try 7.4
7.42 54.76
Too low but very close
Check On a calculator 7.4161984
7.4 rounded to the nearest tenth.
?
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Square roots have special properties that help
you simplify, multiply, and divide them.
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Notice that these properties can be used to
combine quantities under the radical symbol or
separate them for the purpose of simplifying
square-root expressions. A square-root expression
is in simplest form when the radicand has no
perfect-square factors (except 1) and there are
no radicals in the denominator.
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Example 2 Simplifying SquareRoot Expressions
Simplify each expression.
A.
Find a perfect square factor of 32.
Product Property of Square Roots
B.
Quotient Property of Square Roots
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Example 2 Simplifying SquareRoot Expressions
Simplify each expression.
C.
Product Property of Square Roots
D.
Quotient Property of Square Roots
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Check It Out! Example 2
Simplify each expression.
A.
Find a perfect square factor of 48.
Product Property of Square Roots
B.
Quotient Property of Square Roots
Simplify.
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Check It Out! Example 2
Simplify each expression.
C.
Product Property of Square Roots
D.
Quotient Property of Square Roots
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• If a fraction has a denominator that is a
  square root, you can simplify it by rationalizing
  the denominator. To do this, multiply both the
  numerator and denominator by a number that
  produces a perfect square under the radical sign
  in the denominator.

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Example 3A Rationalizing the Denominator
Simplify by rationalizing the denominator.
Multiply by a form of 1.
2
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Example 3B Rationalizing the Denominator
Simplify the expression.
Multiply by a form of 1.
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Check It Out! Example 3a
Simplify by rationalizing the denominator.
Multiply by a form of 1.
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Check It Out! Example 3b
Simplify by rationalizing the denominator.
Multiply by a form of 1.
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Square roots that have the same radicand are
called like radical terms.
To add or subtract square roots, first simplify
each radical term and then combine like radical
terms by adding or subtracting their coefficients.
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Example 4A Adding and Subtracting Square Roots
Add.
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Example 4B Adding and Subtracting Square Roots
Subtract.
Simplify radical terms.
Combine like radical terms.
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Check It Out! Example 4a
Add or subtract.
Combine like radical terms.
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Check It Out! Example 4b
Add or subtract.
Simplify radical terms.
Combine like radical terms.