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Exponents: Basic Rules (page 1 of 5)

Sections: Basics, Negative exponents, Scientific notation, Engineering notation, Fractional exponents

Exponents are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals" sign in (5)(5)(5) = 5³. The "exponent", being 3 in this example, stands for however many times the value is being multiplied. The thing that's being multiplied, being 5 in this example, is called the "base".

This process of using exponents is called "raising to a power", where the exponent is the "power". The expression "5³" is pronounced as "five, raised to the third power" or "five to the third". There are two specially-named powers: "to the second power" is generally pronounced as "squared", and "to the third power" is generally pronounced as "cubed". So "5³" is commonly pronounced as "five cubed".

When we deal with numbers, we usually just simplify; we'd rather deal with "27" than with "3³". But with variables, we need the exponents, because we'd rather deal with "x⁶" than with "xxxxxx".

Exponents have a few rules that we can use for simplifying expressions.

• Simplify (x³)(x⁴)   Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

To simplify this, I can think in terms of what those exponents mean. "To the third" means "multiplying three copies" and "to the fourth" means "multiplying four copies". Using this fact, I can "expand" the two factors, and then work backwards to the simplified form:

(x³)(x⁴) = (xxx)(xxxx) 
          = xxxxxxx 
          = x⁷

Note that x⁷ also equals x⁽³⁺⁴⁾. This demonstrates the first basic exponent rule: Whenever you multiply two terms with the same base, you can add the exponents:

( x ^m ) ( x ⁿ ) = x^{( m + n )}

However, we can NOT simplify (x⁴)(y³), because the bases are different: (x⁴)(y³) = xxxxyyy = (x⁴)(y³). Nothing combines.

• Simplify (x²)⁴

Just as with the previous exercise, I can think in terms of what the exponents mean. The "to the fourth" means that I'm multiplying four copies of x²:

(x²)⁴ = (x²)(x²)(x²)(x²) 
       = (xx)(xx)(xx)(xx) 
       = xxxxxxxx 
       = x⁸

Note that x⁸ also equals x^{( 2×4 )}. This demonstrates the second exponent rule: Whenever you have an exponent expression that is raised to a power, you can multiply the exponent and power:

( x^m ) ⁿ = x ^{m n}

If you have a product inside parentheses, and a power on the parentheses, then the power goes on each element inside. For instance, (xy²)³ = (xy²)(xy²)(xy²) = (xxx)(y²y²y²) = (xxx)(yyyyyy) = x³y⁶ = (x)³(y²)³. Another example would be:

Warning: This rule does NOT work if you have a sum or difference within the parentheses. Exponents, unlike mulitiplication, do NOT "distribute" over addition.

For instance, given (3 + 4)², do NOT succumb to the temptation to say "This equals 3² + 4² = 9 + 16 = 25", because this is wrong. Actually, (3 + 4)² = (7)² = 49, not 25. When in doubt, write out the expression according to the definition of the power. Given (x – 2)², don't try to do this in your head. Instead, write it out: "squared" means "times itself", so (x – 2)² = (x – 2)(x – 2) = xx – 2x – 2x + 4 = x² – 4x + 4.

The mistake of erroneously trying to "distribute" the exponent is most often made when the student is trying to do everything in his head, instead of showing his work. Do things neatly, and you won't be as likely to make this mistake.

There is one other rule that may or may not be covered at this stage:

Anything to the power zero is just "1".

This rule is explained on the next page. In practice, though, this rule means that some exercises may be a lot easier than they may at first appear:

• Simplify [(3x⁴y⁷z¹²)⁵ (–5x⁹y³z⁴)²]⁰

Who cares about that stuff inside the square brackets? I don't, because the zero power on the outside means that the value of the entire thing is just 1.

Division: 

Return to the Lessons Index  | Do the Lessons in Order  |  Get "Purplemath on CD" for offline use  |  Print-friendly page Polynomial Division: Simplification      and Reduction (page 1 of 3) Sections: Simplification and reduction, Polynomial long division There are two cases for dividing polynomials: either the "division" is really just a simplification and you're just reducting a fraction, or else you need to do long polynomial division (which is covered on the next page). Simplify  This is just a simplification problem, because there is only one term in the polynomial that you're dividing by. And, in this case, there is a common factor in the numerator (top) and denominator (bottom), so it's easy to reduce this fraction. There are two ways of proceeding. I can split the division into two fractions, each with only one term on top, and then reduce: ...or else I can factor out the common factor from the top and bottom, and then cancel off: Either way, the answer is the same: x + 2 Simplify  Again, I can solve this in either of two ways: by splitting up the sum and simplifying each fraction separately:   Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved ...or else by taking the common factor out front and canceling it off: Either way, the answer is the same:  3x2 – 5x Note: Most books don't talk about the domain at this point. But if your book does, you will need to note, for the above simplification, that x cannot equal zero. That is, for the simplified form to be completely mathematically equal to the original expression, the solution would need to be "3x2 – 5x, for all x not equal to 0". Simplify   I can split the sum and reduce each fraction separately: The numerator (top) does indeed have a common factor; it's just a rather large one. Since both terms contain the factor "x + 3", then this is a common factor, and may be factored out front: Either way, the answer is the same: x – 2 POLYGONS: Polygons A polygon is a plane shape with straight sides. Is it a Polygon? Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up). Polygon  (straight sides) Not a Polygon  (has a curve) Not a Polygon  (open, not closed) Polygon comes from Greek. Poly- means "many" and -gon means "angle". Types of Polygons Simple or Complex A simple polygon has only one boundary, and it doesn't cross over itself. A complex polygon intersects itself! Simple Polygon (this one's a Pentagon) Complex Polygon (also a Pentagon) Concave or Convex A convex polygon has no angles pointing inwards. More precisely, no internal angles can be more than 180°. If there are any internal angles greater than 180° then it is concave. (Think: concave has a "cave" in it) Convex Concave Regular or Irregular If all angles are equal and all sides are equal, then it is regular, otherwise it is irregular Regular Irregular More Examples Complex Polygon  (a "star polygon", in  this case, a pentagram) Concave Octagon Irregular Hexagon   Names of Polygons     If it is a Regular Polygon... Name Sides Shape Interior Angle Triangle (or Trigon) 3 60° Quadrilateral (or Tetragon) 4 90° Pentagon 5 108° Hexagon 6 120° Heptagon (or Septagon) 7 128.571° Octagon 8 135° Nonagon (or Enneagon) 9 140° Decagon 10 144° Hendecagon (or Undecagon) 11 147.273° Dodecagon 12 150° Triskaidecagon 13   152.308° Tetrakaidecagon 14   154.286° Pentadecagon 15   156° Hexakaidecagon 16   157.5° Heptadecagon 17   158.824° Octakaidecagon 18   160° Enneadecagon 19   161.053° Icosagon 20   162° Triacontagon 30   168° Tetracontagon 40   171° Pentacontagon 50   172.8° Hexacontagon 60   174° Heptacontagon 70   174.857° Octacontagon 80   175.5° Enneacontagon 90   176° Hectagon 100   176.4° Chiliagon 1,000   179.64° Myriagon 10,000   179.964° Megagon 1,000,000   ~180° Googolgon 10100   ~180° n-gon n (n-2) × 180° / n For polygons with 13 or more sides, it is OK (and easier) to write "13-gon", "14-gon" ... "100-gon", etc...