Square and Cube Roots
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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.
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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.
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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.
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Rational number- is any number that can be written as a fraction
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Irrational number- is any number that cannot be written as a fraction
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Example: √ = ±3, which can be written as a fraction or -
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Example √ = a number that cannot be written as fraction, it is a decimal that does not end or repeat
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Square root- is a number that when multiplied by itself gives the number
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Cube root- is a number that when multiplied by itself three times gives the number
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Example √
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√ = 2, is a cube root
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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
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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
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Estimating square roots of non-perfect squares
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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
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Example: Estimate the square root of 12 to the nearest tenth without a calculator
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First, determine between which two perfect squares the number is between, in this case 9 and 16, square
root these
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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
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3.12 = 9.61, this is too small, so you can skip to a higher exponent
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3.3 =10.89, this is still too small
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3.42 = 11.56, this is very close
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3.52 = 12.25, this is too big, so we know that the square root of 12 is between 3.4 and 3.5
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Which of these is closer? 3.5
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So the estimate of the square root of 12 is 3.5
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