Unit 47: Exponents and Radicals

The SAT will expect you to know exponent and radical rules and be able to utilize them.

For example:

If (4a-2b=3), which of the following is equivalent to (frac{16^{2a}}{4^{2b}})?

A) 1
B) (frac72)
C) 64
D) 128

Strategy:

  1. Ask yourself “do they have the same base?” The first step of some exponent problems is to rewrite one of the bases so that it matches another base. (the base is the bottom number that is raised to the exponent).
  2. If you have bases that match, utilize the exponent rules until you’ve fully reduced the expression.

Basic Exponent Rules

  1. When we have the same base and we are multiplying terms, we add the exponents. $$x^2;times;x^3=x^5\xx;times;xxx=xxxxx$$
  2. When we are dividing we subtract the exponents $$frac{x^3}{x^2};=x\frac{xxx}{xx}=x$$
  3. When we have an exponent raised to an exponent, we multiply the exponents.  (Don’t forget the coefficient is also raised to the power!) $${(x}^2)^4=x^8\left(xxright);times;left(xxright);times;left(xxright);times;(xx)=xxxxxxxx\{(2x}^2)^3={8x}^6$$
  4. When we have the same base and exponent, we add (or subtract) the coefficients $${2x}^2+{3x}^2={5x}^2$$

Bonus rule for squared numbers:  remember that the negatives can also be squared to get a positive number.   If you have (x^2=25), could be 5 OR -5.  Don’t forget!

Those are the basic rules.  You must know them to be able to score well on the math section!

Advanced Exponent Rules

There are 3 more advanced methods you may need to know for the SAT.  You might need to do some combination, or all three, on the trickier problems. They are:

  1. With negative exponents, you simply flip the variable to the other side of the divisor and make the exponent positive. $$x^{-2}=frac1{x^2}\frac{x^{-2}y^3}{x^5y^{-1}}=frac{y^1y^3}{x^5x^2}=frac{y^4}{x^7}$$
  2. Changing bases.  When the answer to our crucial question, do they have the same base?, is no, see if you can rewrite one of the bases so that they are the same. You will usually have to rewrite the larger base as the smaller base to an exponent. $$2^x;times{;8}^y=2^x;times;(2^3)^y=2^x;times{;2}^{3y}=2^{x+3y}\frac{3^x}{9^{-2x}}=;3^x;times;9^{2x}=3^x;times;(3^2)^{2x}=;3^x;times;3^{4x}=3^{5x}$$
  3. Switching between radicals and fractional exponents.  Radicals are fractional exponents, and fractional exponents are radicals.  The most common one: $$sqrt x=x^frac12$$
    • Any time you have a fractional exponent, the number on the bottom of the fraction moves out front with the radical, and the numerator of the fraction stays with the base. $$x^frac23=sqrt[3]{x^2}$$
    • You can convert the other way as well: $$sqrt[4]{x^3}=x^frac34$$
    • You may see an improper fraction (numerator is larger than denominator).  Remember we can split this apart. $$8^frac43=8^frac33;times;8^frac13=8^1;times;sqrt[3]8=8;times;2=16$$

If you see an exponent problem on your test that looks totally confusing, don’t despair!  You will be able to apply one or more of these rules to get to the answer.

Now back to our example problem:

Do they have the same base?    (frac{16^{2a}}{4^{2b}}) don’t have the same base. This means we may need to rewrite one of the bases so they are the same.

16 can be written as (4^2). This means our fraction could be written as: (frac{left(4^2right)^{2a}}{4^{2b}}).

According to our exponent rules, (left(4^2right)^{2a}=4^{4a}).

Now we have (frac{4^{4a}}{4^{2b}}=4^{4a-2b}).

Since (4a-2b=3), (4^{4a-2b}=4^3=64).