Which is the value of the expression (StartFraction (10 Superscript 4 Baseline) (5 squared) Over (10 cubed) (5 cubed)) cubed?
Answers
Answer:
Exponents
Overview
Raising a base number x to the power of n => multiplying the base number x by itself n times
Shorthand for multiplying the base number x by itself n times: x Superscript n question-mark Baseline equals y
Exponent (n) notation: x Superscript n
Operations
Procedure to MULTIPLY numbers with exponents, given they have the same base => ADD the exponents
Multiplication example: x squared times x Superscript 5 Baseline equals x Superscript 2 plus 5 Baseline equals x Superscript 7
Procedure to DIVIDE numbers with exponents, given they have the same base => SUBTRACT the exponents
Division example: StartFraction x squared Over x cubed EndFraction equals x Superscript 2 minus 3 Baseline equals x Superscript negative 1
Examples
Exponent example: 3 squared equals 3 times 3 equals 9
Radicals / Roots
Overview
Opposite of exponent - finding the base number x, given y (the result) and the exponent n of the base number - e.g. square root, cube root
To find the base number x: x question-mark Superscript n Baseline equals y
Notations
nth Root notation: RootIndex n StartRoot y EndRoot
Square Root notation: StartRoot y EndRoot
Cube Root notation: RootIndex 3 StartRoot y EndRoot
Examples
Square Root example: x squared equals 9 x equals StartRoot 9 EndRoot equals 3
Logarithms
Overview
The exponent n for a given base number x and result y
To find the exponent n to determine how many times to multiply a base number by itself to get the given result: x Superscript n question-mark Baseline equals y
Log notation: n equals log Subscript x Baseline left-parenthesis y right-parenthesis
Examples
Log example: 5 Superscript n question-mark Baseline equals 25 n equals log Subscript 5 Baseline left-parenthesis 25 right-parenthesis n equals 2
Common Logarithm
The "common logarithm" of a number is an exponential for the base 10
"Common Log notation - default of base 10: log left-parenthesis y right-parenthesis equals log Subscript 10 Baseline left-parenthesis y right-parenthesis
Log example: 10 Superscript n Baseline equals 1000 n equals log Subscript 10 Baseline left-parenthesis 1000 right-parenthesis equals log left-parenthesis 1000 right-parenthesis n equals 3
Natural Logarithm
The "natural logarithm (ln)" of a number is an exponential for the base e, where e is a constant ~2.718
Natural Log notation: ln left-parenthesis y right-parenthesis equals log left-parenthesis e right-parenthesis
Natural Log example: log Subscript e Baseline left-parenthesis 64 right-parenthesis equals ln left-parenthesis 64 right-parenthesis equals 4.1589
Inverse of Natural Log: e Superscript x
Derivative of ln x: StartFraction 1 Over x EndFraction
Properties / Laws
Distributive Property
Allows algebraic expressions in the form of a(b + c) to be reformulated to an equivalent expression
Distributive law: a left-parenthesis b plus c right-parenthesis equals a b plus a c
Khan Academy: Distributive Property
Monomials
Monomial - an algebric expression that contains only one term
Trinomials
Trinomial - an algebric expression that contains only three terms
Factoring
Rewriting a number or term as a product of several smaller factors or values in common
Difference of Squares Pattern
Every polynomial that is a difference of square terms can be factored with this formula: a squared minus b squared equals left-parenthesis a plus b right-parenthesis left-parenthesis a minus b right-parenthesis
Khan Academy: Factoring
Binomials
Overview
Binomial - an algebric expression that contains only two terms
Operations
Difference of Squares Pattern - special products of binomials (applying the Distributive Property twice): left-parenthesis a plus b right-parenthesis left-parenthesis a minus b right-parenthesis equals a left-parenthesis a plus b right-parenthesis minus b left-parenthesis a plus b right-parenthesis equals a squared plus a b minus a b minus b squared equals a squared minus b squared
Squaring binonmials of the form (a+b)^2: left-parenthesis a plus b right-parenthesis squared equals left-parenthesis a plus b right-parenthesis left-parenthesis a plus b right-parenthesis equals a squared plus 2 a b plus b squared
Squaring binonmials of the form (a-b)^2: left-parenthesis a minus b right-parenthesis squared equals left-parenthesis a minus b right-parenthesis left-parenthesis a minus b right-parenthesis equals a squared minus 2 a b plus b squared
Squaring binonmials of the form (ax+b)^2: left-parenthesis x plus a right-parenthesis squared equals left-parenthesis x plus a right-parenthesis left-parenthesis x plus a right-parenthesis equals x squared plus 2 a x plus a squared