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Dividing by 10
Moving the decimal point to the left
Multiplying by 10
Moving the decimal point to the right
When you change the position of the decimal point in a coefficient value
you have to adjust the value of the exponent in order avoid changing the actual value.
decrease the value of the exponent by 1 (dividing by 10)
When moving the decimal point to the right (multiplying by 10)
When moving the decimal point to the left (dividing by 10)
increase the value of the exponent by 1 (multiplying by 10)
To multiply powers of 10:
Step 2. Add the exponents of the factors. The result is the exponent of the product. Of course the base of 10 remains unchanged.
To divide powers of 10:
Step 2. Subtract the exponents of the terms — numerator minus denominator. The result is the exponent of the quotient.
same exponent
Adding and subtracting powers of ten can be a bit more complicated than multiplying and dividing. The main problem is that powers of ten can be added or subtracted only when both terms have the
Are Equal
Powers of ten can be added or subtracted only when their exponents
When the exponents are not the same
rewrite one of the terms so that the exponents are equal
Engineering notation
is a special form of power-of-10 notation where the exponents for the 10s must be 0 or multiples of 3. There must be 1, 2, or 3 digits on the left side of the decimal point.
valid powers of 10 for engineering notation are:
10^3 10^6 10^9 10^ -3 10^ -6 10^ -9 10^0
Valid powers-of-10 for engineering notation
must be multiples of 3 or 0
decrease the power-of-10 exponent by the same number of units
When you move the decimal point in the coefficient to the right
increase the power-of-10 exponent by the same number of units
When you move the decimal point in the coefficient to the left
When you increase the value of the power-of-10 exponent
move the decimal point the same number of units to the left
When you decrease the value of the power-of-10 exponent
move the decimal point the same number of units to the right
Base
the number that is written with an exponent
Density Property
the property that states that between any two real numbers, there is always another real number
exponent
A mathematical notation indicating the number of times a quantity is multiplied by itself
exponential form
a number is in this form when it is written with a base and an exponent
Hypotenuse
in a right triangle, the side opposite the right angle
Irrational Number
A number that cannot be written as a fraction with a numerator and a denominator that are integers. The decimal representation of a rational number either ends or repeats.
Legs
in a right triangle, the sides that include the right angle; in an isosceles triangle, the pair of congruent sides
monomial
a number or a product of numbers and variables with exponents that are whole numbers
Perfect Square
the number that forms when two same numbers are multiplied together
power
a number produced by raising a base to an exponent
Principal Square Root
the nonnegative square root of a number
Pythagorean Theorem
in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs
real number
combonation of rational and irrational numbers
Scientific notation
where a number is written into 2 parts first just the digits (with decimal point after first digit) followed by ×10 to a power
Square Root
when multiplying a number by itself to get a square number
13²
169
14²
196
15²
225
16²
256
17²
289
18²
324
19²
361
20²
400
21²
441
22²
484
23²
529
24²
576
25²
625
26²
676
27²
729
28²
784
29²
841
30²
900
Exponent or power tells you...
how many bases to multiply together
Any base to the first power is just that base.
3¹ = 3
What are 3¹ through 3⁴...
5¹ = 5
What are 5¹ through 5³...
4¹ = 4
What are 4¹ through 4³...
Calculate -3² and (-3)²
-3² = -(3²) = -9
Positive
Counting from the decimal point to the right makes the exponent ___________________________________________
Negative
Counting from the decimal point to the left makes the exponent ___________________________________
Solve for x² = 16
* an even exponent can hide the sign of the base! x = 4 or x = -4
a⁵ × a³ =
a⁵ × a³ = a⁸
y(y⁶) =
y(y⁶) = y¹ × y⁶ = y⁷
2³ × 2ⁿ
2³ × 2ⁿ = 2³⁺ⁿ
a¹⁵/a¹³ =
a¹⁵/a¹³ = a¹⁵⁻¹³ = a²
a⁰ =
a⁰ = 1
a³ / a⁵ =
a³ / a⁵ = a³⁻⁵ = a⁻² OR 1/a²
x²x³
x²⁺³ or x⁵
x²+x²
2x²
x⁵/x³
x⁵⁻³ or x²
x²y²
(xy)²
x³/y³
(x/y)³
(x/y)⁵
x⁵/y⁵
(x³)²
x³*² or x⁶
x³⁻⁷
x³/x⁷ or x⁻⁴ or 1/x⁴
√x²
x
√(a²+a²)
√(2a²)
5√x
3√x + 2√x
√(36*16)
√36 * √16 or 6*4
√(25/4)
√25 / √4 or 5/2
√(a²b)
√a² * √b or a√b
Simplify: 1/√3
(1/√3) * (√3/√3) = (√3/3)
x⁰
1
x⁻²
1/x²
x
x^½
√x
x^¾
⁴√x³
8^²/³
³√8² = ³√64 = 4
Πr²
Area of circle
Circumference
2πr
Sector ratio
(∠/360°) = (Area of sector/ Area of circle)
Arc ratio
(∠/360°) = (Arc Length/ Circumference)
360°
Degrees in a circle
180°
Degrees in a line
slope
(y₂-y₁)/(x₂-x₁)
y=mx+b
Line equation
Positive slope
Moves up
Negative slop
Moves down
horizontal line
Slope of zero
undefined slope
Vertical line
½bh
Area of Triangle
Triangle height
Perpendicular to base. Does not have to be measured in the triangle. Every triangle has 3 heights and 3 bases.
Triangle sides rule
Equal sides are across from equal angles. Largest side is across from the largest angle
isosceles triangle
Triangle with 2 equal sides and 2 equal angles.
equilateral triangle
Triangle with a 90° angle
Third side rule
The third side of any triangle must be between the sum and difference of the other two sides.
Triangle in a semi-circle
A triangle inscribed in a semicircle is a right triangle.
1:1:√2
Side Ratio: 45-45-90 Triangle
Side Ratio: 30-60-90 Triangle
1:2:√3 (2 is hypotenuse)