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Square and Cube Roots Common Core Standard: Use square root and cube root symbols to represent solutions to equations of the formx2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. Common Core Standard: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Common Core Standard: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions. Rational number- is any number that can be written as a fraction Irrational number- is any number that cannot be written as a fraction o Example: √ = ±3, which can be written as a fraction or - o Example √ = a number that cannot be written as fraction, it is a decimal that does not end or repeat Square root- is a number that when multiplied by itself gives the number Cube root- is a number that when multiplied by itself three times gives the number o Example √ o √ = 2, is a cube root o Square roots can be either positive or negative, as a two numbers, which have the same sign, when = ±4, is a square root multiplied are always positive o Cube roots are always positive Perfect Squares and Perfect Cubes- a numbers who always have an exact root the two above examples are perfect square and cube roots Estimating square roots of non-perfect squares o Why – it is important to develop a number sense of non-perfect squares, as these will come up in real world situations, particularly concerning the Pythagorean Theorem o Example: Estimate the square root of 12 to the nearest tenth without a calculator o First, determine between which two perfect squares the number is between, in this case 9 and 16, square root these o Now, using the first of the two perfect squares, determine the square of each tenth, until you get as close as possible to the number 3.12 = 9.61, this is too small, so you can skip to a higher exponent 3.3 =10.89, this is still too small 3.42 = 11.56, this is very close 3.52 = 12.25, this is too big, so we know that the square root of 12 is between 3.4 and 3.5 Which of these is closer? 3.5 So the estimate of the square root of 12 is 3.5 2