The following is a synopsis.  There are three topics under the synopsis:

• multiplication and division of fractions (everyone should understand this very well, and if not, should remediate it right away)
• powers of fractions (if you don't understand this, you should plan to remediate it within the next week or two; when you encounter powers of fractions you don't understand, you should return to this document for examples)
• negative exponents applied to fractions (at this time, this is especially important for University Physics students, who should immediately remediate anything with which they are not very familiar)

If you have questions about any of these examples or rules, use the Question Form to submit your questions.  Be sure to follow the instructions on the form and including a copy of the examples you are asking about, as well as a summary of what you do and do not understand.

The rules for multiplication and division of fractions are simple enough:

(a / b) * (c / d) = (a * c) / (b * d).  For example:

• (5/8) * (3/4) = (5 * 3) / (8 * 4) = 15 / 32
• (J / kg) * (sec / meter) = (J * sec) / (kg * meter).  You might not know what J and kg stand for or how they are related (and that combination of units might not even make sense) but that doesn't prevent you from doing the algebra of the units
• (meters / 1) * (1 / sec) = (meters * 1) / (1 * sec).  Since multiplication by 1 leaves the quantity being multiplied unchanged, this means that meters * 1 = meters, and 1 * sec = sec.

Thus (meters / 1) * (1 / sec) = (meters * 1) / (1 * sec) = meters / sec.

We can turn this around to say that

• meters / sec = (meters / 1) * (1 / sec).

It might not be completely clear why this is useful or important.  For now simply understand how this follows from the rules for multiplying fractions.

(a / b) / (c / d) = (a / b) * (d / c), i.e., to divide by a fraction you multiply by its reciprocal.  For example:

• (5 / 8) / (7 / 11) = (5 / 8 ) * (11 / 7) = (5 * 11) / (8 * 7) = 40 / 77
• (J / kg) / (sec / meter) = (J / kg) * ( meter / sec) = (J * meter) / (kg * sec).

The reciprocal of a is 1 / a.  For example:

• the reciprocal of 5 is 1/5
• the reciprocal of sec is 1/sec
• Note that the reciprocal of a quantity is very different from the quantity itself.  1/5 is clearly not the same as 5.  It might not be as clear that 1/sec is not the same as sec, but this is every bit as true as the obvious statement that 1/5 is not the same as 5.

Things get a little more complicated when we begin talking about powers of fractions, and powers of units:

The fraction a / b can be understood as the ratio of a to b.  We most often use the term 'ratio' when a and b represent numbers (or numerical quantities whose units are identical).

A power of a ratio is equal to the ratio of the powers.  That is, (a / b)^n = a^n / b^n.  For example:

• 5/8 is the ratio of 5 to 8.   (5/8) ^ 2  = 5^2 / 8^2.  That is, the square of the ratio of 5 to 8 is the ratio 5^2 to 8^2, the ratio of the squares.
• This can of course be simplified as (5/8)^2 = 5^2 / 8^2 = 25 / 64.

The expression a / b can also be read as 'a per b'.  This interpretation is more common when a and b express units:

For example the unit expression (km / hr) can be read as 'km per hour'.

The word 'per' always indicates division.

Powers of such units follow the same rules as powers of numerical fractions.  Examples:

• (km / hr) ^ 2 = km^2 / hr^2
• (m^2 / s^2) ^ (1/2) = (m^2)^(1/2) / (s^2)^(1/2) = m / s

Any quantity can be written with denominator 1.  For example:

• the number 5 can be written as 5 / 1
• the quantity sec can be written as sec / 1
• as seen previous the quantity meters / sec is equal to the quantity (meters / 1) * (1 / sec)

Many students some into this course without a good understanding of negative exponents.  So things get a little more challenging when we run into negative exponents:

By the laws of exponents a^-n = 1 / (a^n). 

Following this rule with n = -1 we see that 1 / a can be written as a^(-1), i.e., as the -1 power of a.  For example:

• 1/5 can be written as 5^(-1)
• 1 / sec can be written as (sec)^-1
• 1 / sec^2 is the reciprocal of sec^2, so it can be written as (sec^2)^(-1).

By the laws of exponents, (a^b)^c = a^(b * c), so (sec^2)^(-1) = sec^( 2 * (-1)) = sec^(-2)

• meters / sec can be written as meters * 1/sec = meters * (sec)^-1
• meters / sec^2 can be written as meters * (1 / sec^2) = meters * (sec^(-2)).