Roots and Rational Exponents: a summary

Although addition and multiplication are commutative, exponentiation is not: swapping the value in the base with the value in the exponent will produce a different result (unless, of course, they are the same value):

2^3 \ne 3^2

Therefore, two different inverse functions are needed to solve equations that involve exponential expressions:
– roots, to undo exponents
logarithms, to undo bases

Just as there are many versions of the addition function (one for each number you might wish to add), and many versions of the “logarithm” function (each with a different base), there are many versions of the “root” function: one for each exponent value to be undone.

Notation

The symbol for a root is \sqrt{~~~~}, and is referred to as a “radical“.  It consists of a sort of check mark on the left, followed by a horizontal line, called a “vinculum”, that serves as a grouping symbol (like parentheses) to Continue reading Roots and Rational Exponents: a summary

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Logarithms

Unlike the two most “friendly” arithmetic operations, addition and multiplication, exponentiation is not commutative. You will get a different result if you swap the value in the base with the one in the exponent (unless, of course, they are the same value):

3^2 \ne 2^3

The most significant impact of this lack of commutativity arises when you need to solve an equation that involves exponentiation: two different inverse functions are needed, one to undo the exponent (a root), and a different one to undo the base (a logarithm).

Just as there are many versions of the addition function (adding 2, adding 5, adding 7.23, etc.), and many versions of the “root” function (square roots, cube roots, etc.),  there are also many versions of the “logarithm” function. Each version has a “base”, which corresponds to the base of its inverse exponential expression.

Inverse Functions: Logarithms & Exponentials

Logarithms are labelled with a number that corresponds to the base of the exponential that they undo. For example, the Continue reading Logarithms

Inverse Musings: * and /

Inverse operations and functions are wonderful things. Without them, solving equations would be much more challenging. Yet inverse operations can also be odd beasts. My previous “inverses” post pondered addition and subtraction, which led us (as young students) to expand our initial universe of counting numbers into the integers. Addition and subtraction are operations that only make Continue reading Inverse Musings: * and /

Summary: Algebra

When faced with an algebraic expression or equation, there are only two types of things you can do to it without changing the quantitative relationship that it describes.

Re-write one or more terms in an equivalent form

This can be done to any expression (no equal sign) or equation (with an equal sign) at any time.

There are three common ways in which this is done: by Continue reading Summary: Algebra

Inverse Musings: + and –

Inverse operations and functions are wonderful things. Without them, solving equations would be much more challenging. Yet inverse operations can also be odd beasts. This is the first of several postings on operations/functions and their inverses.

Addition

The first arithmetic operation we all learned was addition. It seems to arise fairly naturally from the counting numbers (1, 2, 3, etc.), and the set of counting numbers is closed when using addition: no matter which two counting Continue reading Inverse Musings: + and –