3NS2.4 Use the inverse relationship between raising to a power and extracting the root of a perfect square integer; for an integer that is not a square, determine without a calculator the two integers between which its square root lies and explain why.Also covered: AF2.2CaliforniaStandards
5Because the area of a square can be expressed using an exponent of 2, a number with an exponent of 2 is said to be squared. You read 32 as “three squared.”3Area = 32The square root of a number is one of the two equal factors of that number. Squaring a nonnegative number and finding the square root of that number are inverse operations.
6Positive real numbers have two square roots, one positive and one negative. The positive square root, or principle square root, is represented by The negative square root is represented by –
7A perfect square is a number whose square roots are integers A perfect square is a number whose square roots are integers. Some examples of perfect squares are shown in the table.
8You can write the square roots of 16 as ±4, which is read as “plus or minus four.” Writing Math
9Additional Example: 1 Finding the Positive and Negative Square Roots of a Number Find the two square roots of each number.A. 4949 = 77 is a square root, since 7 • 7 = 49.49 = –7––7 is also a square root, since –7 • (–7) = 49.The square roots of 49 are ±7.B. 100100 = 1010 is a square root, since 10 • 10 = 100.100 = –10––10 is also a square root, since–10 • (–10) = 100.The square roots of 100 are ±10.
10Check It Out! Example 1Find the two square roots of each number.A. 2525 = 55 is a square root, since 5 • 5 = 25.25 = –5––5 is also a square root, since –5 • (–5) = 25.The square roots of 25 are ±5.B. 144144 = 1212 is a square root, since 12 • 12 = 144.144 = –12––12 is also a square root, since–12 • (–12) = 144.The square roots of 144 are ±12.
11Additional Example 2: Application A square window has an area of 169 square inches. How wide is the window?Find the square root of 169 to find the width of the window. Use the positive square root; a negative length has no meaning.132 = 169So = 13.The window is 13 inches wide.
12Check It Out! Example 2A square shaped kitchen table has an area of 16 square feet. Will it fit through a van door that has a 5 foot wide opening?Find the square root of 16 to find the width of the table. Use the positive square root; a negative length has no meaning.16 = 4So the table is 4 feet wide, which is less than 5 feet, so it will fit through the van door.
13Additional Example 3: Finding the Square Root of a Monomial Simplify the expression.A.144c2144c2 = (12c)2Write the monomial as a square.= 12|c|Use the absolute-value symbol.B.z6Write the monomial as a square: z6 = (z3)2z6 = (z3)2= |z3|Use the absolute-value symbol.
14Additional Example 3: Finding the Square Root of a Monomial Simplify the expression.C.100n4100n4 = (10n2)2Write the monomial as a square.= 10n210n2 is nonnegative for all values of n. The absolute-value symbol is not needed.
15Check It Out! Example 3Simplify the expression.A.121r2121r2 = (11r)2Write the monomial as a square.= 11|r|Use the absolute-value symbol.B.p8Write the monomial as a square: p8 = (p4)2p8 = (p4)2= |p4|Use the absolute-value symbol.
16Check It Out! Example 3Simplify the expression.C.81m481m4 = (9m2)2Write the monomial as a square.= 9m29m2 is nonnegative for all values of m. The absolute-value symbol is not needed.
17Lesson QuizFind the two square roots of each number.Simplify each expression.p z8±12±507|p3|z45. Ms. Estefan wants to put a fence around 3 sides of a square garden that has an area of 225 ft2. How much fencing does she need?45 ft