You may be thinking that multiplying by zero is no big deal. After all, zero times anything is zero, right? Yes, and that’s the big deal. You can use the multiplication property of zero when solving equations. If you can factor an equation – in other words, write it as the product of two or more multipliers – you can apply the multiplication property of zero to solve the equation. The multiplication property of zero states that
If the product of
, at least one of the factors has to represent the number 0.
The only way the product of two or more values can be zero is for at least one of the values to actually be zero. If you multiply (16)(467)(11)(9)(0), the result is 0. It doesn’t really matter what the other numbers are – the zero always wins.
The reason this property is so useful when solving equations is that if you want to solve the equation x7 – 16x5 + 5x4 – 80x2 = 0, for instance, you need the numbers that replace the x’s to make the equation a true statement. This particular equation factors into x2(x3 + 5)(x – 4)(x + 4) = 0. The product of the four factors shown here is zero. The only way the product can be zero is if one or more of the factors is zero. For instance, if x = 4, the third factor is zero, and the whole product is zero. Also, if x is zero, the whole product is zero. (Head to Chapters 3 and 8 for more info on factoring and using the multiplication property of zero to solve equations.)
The birth of negative numbers
In the early days of algebra, negative numbers weren’t an accepted entity. Mathematicians had a hard time explaining exactly what the numbers illustrated; it was too tough to come up with concrete examples. One of the first mathematicians to accept negative numbers was Fibonacci, an Italian mathematician. When he was working on a financial problem, he saw that he needed what amounted to a negative number to finish the problem. He described it as a loss and proclaimed, “I have shown this to be insoluble unless it is conceded that the man had a debt.”
Expounding on Exponential Rules
Several hundred years ago, mathematicians introduced powers of variables and numbers called exponents. The use of exponents wasn’t immediately popular, however. Scholars around the world had to be convinced; eventually, the quick, slick notation of exponents won over, and we benefit from the use today. Instead of writing xxxxxxxx, you use the exponent 8 by writing x8. This form is easier to read and much quicker.
The expression an is an exponential expression with a base of a and an exponent of n. The n tells you how many times you multiply the a times itself.
You use radicals to show roots. When you see
, you know that you’re looking for the number that multiplies itself to give you 16. The answer? Four, of course. If you put a small superscript in front of the radical, you denote a cube root, a fourth root, and so on. For instance,
, because the number 3 multiplied by itself four times is 81. You can also replace radicals with fractional exponents – terms that make them easier to combine. This use of exponents is very systematic and workable – thanks to the mathematicians that came before us.
Multiplying and dividing exponents
When two numbers or variables have the same base, you can multiply or divide those numbers or variables by adding or subtracting their exponents:
✔
: When multiplying numbers with the same base, you add the exponents.
✔
: When dividing numbers with the same base, you subtract the exponents (numerator – denominator). And, in this case,
.
Also, recall that a0 = 1. Again,
. To multiply
, for example, you add the exponents: x4 + 5 = x9. When dividing x8 by x5, you subtract the exponents:
. You must be sure that the bases of the expressions are the same. You can multiply 32 and 34, but you can’t use the rule of exponents when multiplying 32 and 43.
Getting to the roots of exponents
Radical expressions – such as square roots, cube roots, fourth roots, and so on – appear with a radical to show the root. Another way you can write these values is by using fractional exponents. You’ll have an easier time combining variables with the same base if they have fractional exponents in place of radical forms:
✔
: The root goes in the denominator of the fractional exponent.
✔
: The root goes in the denominator of the fractional exponent, and the power goes in the numerator.
So, you can say
and so on, along with
.
To simplify a radical expression such as
, you change the radicals to exponents and apply the rules for multiplication and division of values with the same base (see the previous section):
You can raise numbers or variables with exponents to higher powers or reduce them to lower powers by taking roots. When raising a power to a power, you multiply the exponents. When taking the root of a power, you divide the exponents:
✔ (am)n = am · n: Raise a power to a power by multiplying the exponents.
✔
: Reduce the power when taking a root by dividing the exponents.
The second rule may look familiar – it’s one of the rules that govern changing from radicals to fractional exponents (see Chapter 4 for more on dealing with radicals and fractional exponents).
Here’s an example of how you apply the two rules when simplifying an expression:
You use negative exponents to indicate that a number or variable belongs in the denominator of the term:
and
Writing variables with negative exponents allows you to combine those variables with other factors that share the same base. For instance, if you have the expression
, you can rewrite the fractions by using negative exponents and then simplify by using
