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Integer Exponents

An exponent is shorthand for repeated multiplication, and every law below is that one fact read back in a different direction. Almost none of it has to be memorised separately: if a rule ever looks doubtful, write the powers out long-hand and count. This section is the floor the next two stand on, because rational exponents and radicals obey exactly these laws.

Class Notes — Chapter 1, Section 1.1

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The laws
Every law assumes the base is non-zero wherever a zero would divide. Read each one both ways — left to right expands, right to left collects, and you need both directions.
  • Product

    Same base multiplied: add the exponents. Different bases never combine.

  • Quotient

    Same base divided: top exponent minus bottom exponent.

  • Power of a power

    A power raised to a power multiplies the exponents. This is the one most often confused with the product rule.

  • Power of a product

    The exponent reaches every factor inside the bracket — factors only, never terms added together.

  • Power of a quotient

    Top and bottom each take the exponent.

  • Zero exponent

    Anything non-zero raised to zero is 1, however big and ugly the base looks.

  • Negative exponent

    A negative exponent is an instruction to move, not a negative number. The sign of the answer is untouched.

  • Negative power of a fraction

    Flip the fraction and the exponent turns positive. Much faster than distributing first.

  • Crossing the bar

    A factor crosses the fraction bar and its exponent changes sign. This is the workhorse for finishing an answer off.

Only the last three are genuinely new — they are the first six extended to negative exponents, which is also why has to be 1: is both 1 and , so the two must agree.

How to simplify one of these
The simplify problems all yield to the same order of moves. Working out of order is what turns a two-line problem into half a page.

  1. Clear the innermost brackets first. If the whole expression sits inside a bracket raised to a power, tidy the inside until it is one power per base, and apply the outer exponent once at the end. Distributing it early multiplies the number of exponents you have to track.

  2. Collect each base separately. Gather all the 's, all the 's, all the 's. Add exponents on top, add exponents on the bottom, then subtract bottom from top. Bases never mix.

  3. Evaluate numbers, leave letters symbolic. A numeric coefficient like should become an actual number, . A letter stays as a power.

  4. Move everything negative across the bar. Last step, always. Any factor with a negative exponent crosses the bar and its exponent flips sign. The answer is finished when no exponent is negative and no base appears twice.

Worked examples
Each line names the move that produced it, so a solution reads as a list of decisions rather than a wall of algebra.

True or false
These are not busywork. Each is a rule that does not exist, written out so it can be recognised on sight. Say why it fails, then give the corrected version.

  • A negative exponent only moves the factor it is attached to. Writing leaves the 6 upstairs, which is a different number: , not .

  • A power of a power multiplies; only a product of powers adds. is the one that gives .

  • Both exponents are multiplied by the 2, not by each other. The was handled correctly and the was not, which is what makes this one easy to nod through.

  • Powers distribute over multiplication, never over addition. One numeric test settles it: but .

  • This one is correct: nested powers multiply all the way down, . Not every statement in a true-or-false set is false, so check rather than assume.

Practice
Work each one on paper before opening the answer.

Type an answer and it gets marked. What you get right is kept on this device.