copyright © 2011 pearson education, inc. publishing as prentice hall. 7.3 square roots and the...
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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
7.3
Square Roots and the Pythagorean
Theorem
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 22
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
The square of a number is the number times itself.
The square of 6 is 36 because 62 = 36.
The square of –6 is also 36 because
The Square of a Number
(–6)2 = (–6) (–6) = 36.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
A square root of 36 is A square root of 36 is 66 because because 62 = 36.
A square root of 36 is also A square root of 36 is also –66 because ( because (–6)2 = 36.
Square Root of a Number
We use the symbol , called a radical sign, to
indicate the positive square root of a nonnegative
number.
because 4because 422 = 16 and 4 is positive. = 16 and 4 is positive. 16 4
25 5 because 5because 522 = 25 and 5 is positive. = 25 and 5 is positive.
The reverse process of squaring is finding a square root.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Square Root of a Number
The square root, , of a positive number a is the
positive number b whose square is a. In symbols,
2 if .a b b a
29 3 because 3 9.
Also, 0 0.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Remember that the radical sign is
used to indicate the positive square root
of a nonnegative number.
Helpful Hint
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 66
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Numbers like are
called perfect squares because their square
root is a whole number or a fraction.
Perfect Squares
1 4, 36, , and 14 25
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 77
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
A square root such as cannot be
written as a whole number or a fraction
since 6 is not a perfect square. It can be
approximated by estimating, by using a
table, or by using a calculator.
Approximating Square Roots
6
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 88
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
One important application of square roots has to do with right triangles.
A right triangle is a triangle in which one of the angles is a right angle or measures 90º (degrees).
TheThe hypotenuse of a right triangle is the side opposite the of a right triangle is the side opposite the right angle.right angle.
hypotenuseleg
leg
TheThe legs of a right triangle are the other two sides.of a right triangle are the other two sides.
Right Triangles
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 99
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Pythagorean Theorem
If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then
In other words,
cc aa
bb
(leg)2 + (other leg)2 = (hypotenuse)2.
2 2 2a b c