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Chapter 7 Exponents and Exponential Functions

Zero as an Exponent

For every nonzero number a , a 0 = 1 . eh comma , eh to the , equals 1 .

Negative Exponent

For every nonzero number a and integer n, a − n = 1 a n . eh super negative n end super , equals , fraction 1 , over eh to the n end fraction , .

Scientific Notation

A number in scientific notation is written as the product of two factors in the form a × 10 n , eh times , 10 to the n , comma  where n is an integer and 1 ≤ a < 10 . 1 less than or equal to eh less than 10 .

Multiplying Powers With the Same Base

For every nonzero number a and integers m  and  n , a m · a n = a m + n . m , and , n comma , eh to the m , middle dot , eh to the n , equals . eh super m plus n end super . .

Dividing Powers With the Same Base

For every nonzero number a and integers m and n , a m a n = a m − n . n comma . fraction eh to the m , over eh to the n end fraction . equals . eh super m minus n end super . .

Raising a Power to a Power

For every nonzero number a and integers m and n , ( a m ) n = a m n . n comma . open , eh to the m , close to the n . equals , eh super m n end super , .

Raising a Product to a Power

For every nonzero number a and b and integer n , ( a b ) n = a n b n . n comma . open eh b close to the n . equals , eh to the n , b to the n , .

Raising a Quotient to a Power

For every nonzero number a and b and integer n, ( a b ) n = a n b n . open , eh over b , close to the n . equals . fraction eh to the n , over b to the n end fraction . .

Geometric Sequence

The form for the rule of a geometric sequence is A ( n ) = a · r n − 1 , eh open n close equals eh middle dot r , n super negative 1 end super , comma  where A(n) is the nth term, a is the first term, n is the term number, and r is the common ratio.

Exponential Growth and Decay

An exponential function has the form y = a · b x , y equals eh middle dot , b to the x , comma  where a is a nonzero constant, b is greater than 0 and not equal to 1, and x is a real number.

• The function y = a · b x , y equals eh middle dot , b to the x , comma  where b is the growth factor, models exponential growth for a > 0 eh greater than 0  and b > 1 . b greater than 1 .
• The function y = a · b x , y equals eh middle dot , b to the x , comma  where b is the decay factor, models exponential decay for a > 0 eh greater than 0  and 0 < b < 1 . 0 less than b less than 1 .

Chapter 8 Polynomials and Factoring

Factoring Special Cases

For every nonzero number a and b:

a 2 − b 2 = ( a + b ) ( a − b ) eh squared , minus , b squared , equals open eh plus b close open eh minus b close

a 2 + 2 a b + b 2 = ( a + b ) ( a + b ) = ( a + b ) 2 eh squared , plus 2 eh b plus , b squared , equals open eh plus b close open eh plus b close equals . open eh plus b close squared

a 2 − 2 a b + b 2 = ( a − b ) ( a − b ) = ( a − b ) 2 eh squared , minus 2 eh b plus , b squared , equals open eh minus b close open eh minus b close equals . open eh minus b close squared

Chapter 9 Quadratic Functions and Equations

Graph of a Quadratic Function

The graph of y = a x 2 + b x + c , y equals , eh x squared , plus b x plus c comma  where a ≠ 0 , eh not equal to 0 comma  has the line x = − b 2 a x equals . fraction negative b , over 2 eh end fraction  as its axis of symmetry. The x-coordinate of the vertex is − b 2 a . fraction negative b , over 2 eh end fraction . .

Zero-Product Property

For every real number a and b, if a b = 0 , eh b equals 0 comma  then a = 0 eh equals 0  or b = 0 . b equals 0 .

Quadratic Formula

If a x 2 + b x + c = 0 , eh x squared , plus b x plus c equals 0 comma  where a ≠ 0 , eh not equal to 0 comma  then x = − b ± b 2 − 4 a c 2 a . x equals . fraction negative b plus minus . square root of b squared , minus 4 eh c end root , over 2 eh end fraction . .

Property of the Discriminant

For the quadratic equation a x 2 + b x + c = 0 , eh x squared , plus b x plus c equals 0 comma  where a ≠ 0 , eh not equal to 0 comma  the value of the discriminant b 2 − 4 a c b squared , minus 4 eh c  tells you the number of solutions.

• If b 2 − 4 a c > 0 , b squared , minus 4 eh c greater than 0 comma  there are two real solutions.
• If b 2 − 4 a c = 0 , b squared , minus 4 eh c equals 0 comma  there is one real solution.
• If b 2 − 4 a c < 0 , b squared , minus 4 eh c less than 0 comma  there are no real solutions.

Chapter 10 Radical Expressions and Equations

The Pythagorean Theorem

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse: a 2 + b 2 = c 2 . eh squared , plus , b squared , equals , c squared , .

The Converse of the Pythagorean Theorem

If a triangle has sides of lengths a, b, and c, and a 2 + b 2 = c 2 , eh squared , plus , b squared , equals , c squared , comma  then the triangle is a right triangle with hypotenuse of length c.

Multiplication Property of Square Roots

For every number a ≥ 0 eh greater than or equal to 0  and b ≥ 0 , a b = a ⋅ b . b greater than or equal to 0 comma . square root of eh b end root , equals square root of eh dot square root of b .

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