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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
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
so it lies between 2 and 3.
Ex 1: Estimate
to the nearest tenth.
Find the two perfect squares that
27 lies between.
Find the two integers that
lies between
Because 27 is closer to 25 than to 36,
Try 5.2: 5.22 = 27.04
5.12 = 26.01
Too high, try 5.1.
Too low
Because 27 is closer to 27.04 than 26.01,
than to 5.1.
Check On a calculator
to the nearest tenth. 
is close to 5 than to 6.
is closer to 5.2
≈ 5.1961524 ≈ 5.1 rounded
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.
Ex 2: Simplify each expression.
Find a perfect square
factor of 32.
Product Property of
Square Roots
Product Property of
Square Roots
Quotient Property of
Square Roots
Quotient Property
of Square Roots
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
Ex 3: Simplify by rationalizing the denominator.
Multiply by a
form of 1.
Multiply by a
form of 1.
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.
Ex 4: Adding and Subtracting Square Roots
radical terms.
Combine like
radical terms.