Why a fractional power is not a division, shown on the curve.
Ask for 16 to the power a half before you slide anything.
Someone will say 8. It is 4, and on the curve the half sits between 1 and 16 but nothing like halfway along, because the scale is multiplicative.
Then slide negative. The curve never crosses the axis, which settles the other standing error: a negative exponent gives a reciprocal, never a negative value.
Both orders are correct, but taking the root first keeps the numbers small: 82/3 is (³√8)² = 4, where the other order needs the cube root of 64.
Nothing new is being learned here. These are the same three laws from 1.5 with fractions in place of integers, and saying so reduces the perceived load considerably.
| 1. 161/2 | 4. |
| 2. 82/3 | 4. |
| 3. 16−1/2 | 0.25. |
1 markBottom as the root, top as the power.
1 markNegative powers as reciprocals, with a positive answer.
1 markExact answers left exact when the question asks for them.
| They give | What it means |
|---|---|
| 8 (Q1) | Divided by 2. The exponent is not a multiplier, and this is the whole point of the page. |
| 5.33 (Q2) | Multiplied 8 by two thirds. |
| 2 (Q2) | Took the cube root and stopped. |
| −4 (Q3) | Read the minus as a sign on the answer rather than a reciprocal. |
| 4 (Q3) | Ignored the minus altogether. |
"Why does x^(1/2) mean a square root?" Because the laws demand it: x^(1/2) times x^(1/2) is x^1, and the only thing that multiplies by itself to give x is the square root.
"Does the order matter?" No, but the arithmetic does. Root first is almost always easier by hand.
"Can the base be negative?" Not safely at this level. (−8)^(1/3) has an answer but (−8)^(1/2) does not, so the courses keep bases positive.
| What is happening | |
|---|---|
| 1 | 16 to the power a half. Reveal on the curve. |
| 2 | The rule, with root-first worked twice. |
| 3 | Negative and zero exponents. |
| 4 | The three laws with fractional exponents. |
| 5 | Algebraic simplification, which is where this is examined. |
Do not teach this as a new topic. It is 1.5 with fractions, and framing it as new makes it harder than it is.
Do not use negative bases, even to be interesting. It opens a door the course deliberately keeps shut.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.