Algebra II For Dummies. Sterling Mary Jane

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multiplier (or coefficient) of the variable.

      Ordering Your Operations

      When mathematicians switched from words to symbols to describe mathematical processes, their goal was to make dealing with problems as simple as possible; however, at the same time, they wanted everyone to know what was meant by an expression and for everyone to get the same answer to the same problem. Along with the special notation came a special set of rules on how to handle more than one operation in an expression. For instance, if you do the problem

, you have to decide when to add, subtract, multiply, divide, take the root, and deal with the exponent.

      

The order of operations dictates that you follow this sequence:

      1. Raise to powers or find roots.

      2. Multiply or divide.

      3. Add or subtract.

      

If you have to perform more than one operation from the same level, work those operations moving from left to right. If any grouping symbols appear, perform the operation inside the grouping symbols first.

      So, to simplify

, follow the order of operations:

      1. The radical acts like a grouping symbol, so you subtract what’s in the radical first:

.

      2. Raise the power and find the root:

.

      3. Multiply and divide, working from left to right: 4 + 9 – 30 + 4 + 7.

      4. Add and subtract, moving from left to right: 4 + 9 – 30 + 4 + 7 = –6.

      Zeroing in on the Multiplication Property of Zero

      

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: x^{4 + 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

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