➕ Algebra 1 · Exponents and Polynomials
Zero and Negative Exponents: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Splitting the cost of a gift
Six friends buy a gift and split the cost evenly. Each friend pays one sixth of the total, which is 6−1 of it. Add more friends and everyone’s share gets smaller. If only one friend pays, that friend covers all of it: 1−1 = 1.
How it connects: 6−1 = 1/6, so a negative exponent gives a share, a fraction of the whole. Each share is a positive amount, because nobody pays a negative price. Limit: this picture shows the exponent −1. Larger negative exponents mean splitting a share again and again.
Halving a pizza
A whole pizza counts as 1. Cut it in half and each piece is 1/2 of a pizza. Cut every piece in half again and each is 1/4. Cut again and each is 1/8. Written as powers of 2: 20 = 1, then 2−1 = 1/2, then 2−2 = 1/4, then 2−3 = 1/8.
How it connects: Each cut divides by 2, which lowers the exponent by 1. With a base bigger than 1, a negative exponent gives something smaller, never a negative amount of pizza. Limit: halving matches the base 2. For another base, each step down divides by that base, and dividing by a fraction makes a number bigger.
Octaves
On a piano, the A above middle C vibrates 440 times per second in standard tuning. The next A up has double the frequency, and the next A down has half. Going down 1 octave gives 440 · 2−1 = 220 vibrations per second. Going down 2 octaves gives 440 · 2−2 = 110.
How it connects: Moving n octaves multiplies the frequency by 2n. A negative n means moving down, which divides by 2 each time. The frequency gets lower, but it never becomes negative.
The combo multiplier
Imagine a game where every hit in a row multiplies your bonus by 4. After 1 hit the multiplier is 41 = 4, after 2 hits it is 42 = 16. With zero hits in a row, the multiplier is 40 = 1: your points stay as they are.
How it connects: Multiplying by 4 zero times leaves your points unchanged, so the multiplier is 1, not 0. If it were 0, a player with no combo would lose every point. This is one way to see why x0 = 1. Limit: this picture explains the zero exponent only. The main reason is that it keeps the quotient rule working.
Turn the tables
To “turn the tables” on someone means to reverse a situation, so that the side that was behind is now ahead: the two positions swap.
How it connects: A negative exponent turns the tables on a fraction: top and bottom swap. x−3 becomes 1/x3, and (5/7)−1 becomes 7/5. Limit: the swap works for factors, not for terms that are added, and it changes the exponent’s sign, not the sign of the number.
Running pond algae backward in time
Imagine a patch of pond algae that triples in area every day, and covers 1 square meter today. In 2 days it will cover 32 = 9 square meters. Two days ago it covered 3−2 = 1/9 of a square meter, because each day further back divides the area by 3.
How it connects: Think of the exponent as days from today: 30 = 1 is today, and a negative exponent means earlier days. Earlier means smaller, never negative. Limit: this is an imaginary patch with a perfectly steady growth rate.
Number one
Why do all the nonzero numbers love the power of zero? Because it makes every one of them number one.
How it connects: x0 = 1 for every nonzero x. The catch is in the word nonzero: 00 is left undefined in this course.
Memory help
Flip it and you’re done
Zero exponent? Answer is one. Negative exponent? Flip it and you’re done. “Flip” means cross the fraction bar and make the exponent positive. The zero rule is for nonzero bases only.
The elevator
Picture a fraction as a two-story building: the numerator is upstairs and the denominator is downstairs. A factor with a negative exponent is on the wrong floor. It rides the elevator to the other floor, and its exponent turns positive when it arrives. x−3 upstairs rides down and becomes 1/x3, and 1/x−3 rides up and becomes x3. Only the factor with the negative exponent rides. In 3x−2, the 3 stays upstairs.
Draw the ladder
Draw a ladder with the rungs 32, 31, 30, 3−1, 3−2 and the values 9, 3, 1, 1/3, 1/9. Each step down divides by 3. The 30 rung is always 1, because it is one step below the base itself: x1 = x, and x ÷ x = 1. If you forget a rule, redraw the ladder with any nonzero base.
A negative exponent is not a negative number
2−3 is not −8 and not −6. It is 1/8, which is positive. The minus sign in an exponent tells you to flip, not to negate. A power with a negative exponent comes out negative only when its base is negative and the exponent is odd: (−2)−3 = −1/8. A minus sign in front is a separate step: −2−3 = −(1/8).
The exponent hugs the left
An exponent hugs only the thing directly to its left: a number, a letter, or a whole parenthesized group. In 4x−1 only the x flips (4/x), but in (4x)−1 both do (1/(4x)). In −20 only the 2 gets the zero exponent, so the answer is −1.