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Algebra II For Dummies. Sterling Mary Jane

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: 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):

Raising or lowering the roof with exponents

      

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 the rules for multiplying factors with the same base (see “Multiplying and dividing exponents”):

Implementing Factoring Techniques

      When you factor an algebraic expression, you rewrite the sums and differences of the terms as a product. For instance, you write the three terms x2 – x – 42 in factored form as (x – 7)(x + 6). The expression changes from three terms to one big, multiplied-together term. You can factor two terms, three terms, four terms, and so on for many different purposes. The factorization comes in handy when you set the factored forms equal to zero to solve an equation. Factored numerators and denominators in fractions also make it possible to reduce the fractions.

      You can think of factoring as the opposite of distributing. You have good reasons to distribute or multiply through by a value – the process allows you to combine like terms and simplify expressions. Factoring out a common factor also has its purposes for solving equations and combining fractions. The different formats are equivalent – they just have different uses.

Factoring two terms

      

When an algebraic expression has two terms, you have four different choices for its factorization – if you can factor the expression at all. If you try the following four methods and none of them work, you can stop your attempt; you just can’t factor the expression:

      

In general, you check for a greatest common factor before attempting any of the other methods. By taking out the common factor, you often make the numbers smaller and more manageable, which helps you see clearly whether any other factoring is necessary.

      To factor the expression 6x4 – 6x, for example, you first factor out the common factor, 6x, and then you use the pattern for the difference of two perfect cubes:

      

A quadratic trinomial is a three-term polynomial with a term raised to the second power. When you see something like x2 + x + 1 (as in this case), you immediately run through the possibilities of factoring it into the product of two binomials. In this case, you can just stop. These trinomials that crop up when factoring cubes just don’t cooperate.

      Keeping in mind my tip to start a problem off by looking for the greatest common factor, look at the example expression 48x3y2 – 300x3. When you factor the expression, you first divide out the common factor, 12x3, to get 12x3(4y2 – 25). You then factor the difference of perfect squares in the parentheses: 12x3(4y2 – 25) = 12x3(2y – 5)(2y + 5).

      Here’s one more: The expression z4 – 81 is the difference of two perfect squares. When you factor it, you get z4 – 81 = (z2 – 9)(z2 + 9). Notice that the first factor is also the difference of two squares – you can factor again. The second term, however, is the sum of squares – you can’t factor it. With perfect cubes, you can factor both differences and sums, but not with the squares. So, the factorization of z4 – 81 is (z – 3)(z + 3)(z2 + 9).

Taking on three terms

      

When a quadratic expression has three terms, making it a trinomial, you have two different ways to factor it. One method is factoring out a greatest common factor, and the other is finding two binomials whose product is identical to those three terms:

      You can often spot the greatest common factor with ease; you see a multiple of some number or variable in each term. With the product of two binomials, you either find the product or become satisfied that it doesn’t exist.

      For example, you can perform the factorization of 6x3 – 15x2y + 24xy2 by dividing each term by the common factor, 3x: 6x3 – 15x2y + 24xy2 = 3x(2x2 – 5xy + 8y2).

      

You want to look for the common factor first; it’s usually easier to factor expressions when the numbers are smaller. In the previous example, all you can do is pull out that common factor – the trinomial is prime (you can’t factor it any more).

      Trinomials that factor into the product of two binomials have related powers on the variables in two of the terms. The relationship between the powers is that one is twice the other. When factoring a trinomial into the product of two binomials, you first look to see if you have a special product: a perfect square trinomial. If you

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