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multiple
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The product of a specified number and some integer.
For example, 3, 12 and 90 are all multiples of 3. 4 is not a multiple of 3 because there is no integer that can be multiplied by 3 and yield 4
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An integer is divisible by 2 if...
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...if its last digit is divisible by 2
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An integer is divisible by 3 if...
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...if its digits sum to a multiple of 3
6,930 is a multiple of 3 because 6+9+3+0=18 which is a multiple of 3
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An integer is divisible by 4 if...
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...if its last two digits are a multiple of 4
4,716/4 = 1179
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An integer is divisible by 5 if...
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...if its last digit is 0 or 5
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An integer is divisible by 6 if...
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...if it divisible by 2 and 3
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An integer is divisible by 9 if...
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...if its digits sum to a multiple of 9
6,930...6+9+3+0=18, which is a multiple of 9
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factors (aka divisors)
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The factors of an integer are the positive integers by which it is evenly divisible. 36 has 9 factors: 1, 2, 3, 4, 6, 9, 12, 18, and 36
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5% as a decimal and fraction
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0.05 and 1/20
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12.5% as a decimal and fraction
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0.125 and 1/8
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20% as a decimal and fraction
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0.2 and 1/5
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33 1/3% as a decimal and fraction
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.3333 and 1/3
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10% as a decimal and fraction
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.1 and 1/10
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16 2/3 % as a decimal and fraction
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0.16666 and 1/6
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percent formula
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PART/WHOLE x 100 = PERCENT
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percent increase (or decrease)
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amount of increase/original whole X 100
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distance =
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rate x time
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to multiple powers with the same base...
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add the exponents and keep the base
7^3 x 7^5 = 7^8
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to divide powers with the same base...
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subtract the exponents and keep the base the same
4^5/4^2 = 4^3
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to multiple powers (or raise a power to a power)
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multiply the exponents
7^2(^3) = 7^6
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a negative number raised to an even power...
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...yields a positive result (-1)^2 = 1
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a negative number raised to an odd power...
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...yields a negative result
(-1)^57 = -1
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raising a fraction between zero and 1 to a power...
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...yields a smaller result
(1/2)^2 = 1/4
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what happens when an exponent is negative?
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Take the reciprocal of the base and change the sign of the exponent
(2)^-2 = 1^2/2^2 = 1/4
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what happens when an exponent is negative?
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Take the reciprocal of the base and change the sign of the exponent
(2)^-2 = 1^2/2^2 = 1/4
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raising a fraction between zero and 1 to a power...
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...yields a smaller result
(1/2)^2 = 1/4
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what happens when an exponent is negative?
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Take the reciprocal of the base and change the sign of the exponent
(2)^-2 = 1^2/2^2 = 1/4
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Express 9^1/2 as a radical
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= the square root of 9 = 3
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Express 8^1/3 as a radical
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= the cube root of 8 = 2
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to simplify a radical...
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factor out the perfect squares and move them to the front of the radical sign. For example, the square root of 50 = 5 square root 2
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when can radicals be added and subtracted?
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only when the number under the radical is the same! 6radical7 + 2radical7 = 8radical7
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to multiply radicals...
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...multiply the numbers under the signs and then put a single radical sign over them the new number
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to divide radicals...
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...divide the two numbers in question and then put them under a single radical
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if multiplying or dividing an inequality by a negative number...
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REVERSE the inequality sign
-3x < 6 = x > 2
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supplementary angles
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two angles are supplementary if their measures sum to 180
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complementary angles
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two angles are complementary if their measures sum to 90
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adjacent angles
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angles that are adjacent (next to each other) are supplementary because they lie along a straight line
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vertical angles
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two angles that are not adjacent to each other are opposite, or vertical, and are equal in measure
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perimeter of a triangle
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the sum of the lengths of all three sides
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area of a triangle
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area of a triangle = 1/2(base)(height)
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isosceles triangles
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an isosceles triangle has two equal sides and the angles opposite these sides are equal as well
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equilateral triangle
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all three sides of an equilateral triangle are equal and the interior angles equal 60
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right triangles
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triangles with one interior angle of 90. The hypotenuse lies opposite the right angle. The other two sides are legs. leg^2 + leg^2 = hypotenuse^2
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pythagorean triplets (2)
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3:4:5 (leg:3 leg:4 hypotenuse:5)
and
5:12:13
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isosceles right triangles
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angles = 45, 45, and 90
the ratio of sides is always 1:1:root2
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30-60-90 right triangles
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the ratio of sides is always 1:root3:2
paired as follows...the side opposite the 30 degree angle is 1, etc.
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define: quadrilateral
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a four sided polygon where the four interior angles add up to 36, regardless of the quadrilateral's shape
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define: parallelogram
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a parallelogram has two pairs of equal sides. Opposite angles are equal. Consecutive angles add up to 180
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define: rectangle
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a quadrilateral with four right angles. Opposite sides are equal.
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perimeter of a rectangle
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perimeter = 2(length + width)
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