At a glance. One lesson of about 60 minutes. It assumes the laws of indices, zero and negative exponents and square and cube roots. Students evaluate and simplify with fractional indices and solve simple equations with an unknown exponent.
Calculator: GDC optional. Everything can be done by hand. The optional section shows students who have a GDC how to enter a fractional exponent in brackets, and why the decimal is not the exact answer.
What students will learn
Know
- x to the power 1/n is the nth root of x, for x > 0.
- x to the power m/n is the nth root of x raised to the power m, in either order.
- The laws of indices hold for fractional indices.
Understand
- A fractional index is forced by the laws: the nth power of x to the power 1/n must be x.
- Roots and powers are one system, so any root can be written as a power.
- The definition needs x > 0 so that the value does not depend on how the fraction is written.
Do
- Evaluate expressions with positive and negative fractional indices without a calculator.
- Simplify products and powers of expressions with fractional indices.
- Solve equations with an unknown index by writing both sides as the same base.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: evaluating and simplifying with fractional indices (questions 1 to 9).
- Criterion B. Investigating patterns: the first figure and the what-if. Students generalize x^(m/n) and discover why x must be positive.
- Criterion C. Communicating: writing a root as a power and justifying each step with a law of indices (worked examples 1 to 3).
- Criterion D. Applying mathematics in real-life contexts: growth over a fraction of a time period, as in the hook and question 7, where the factor for part of a period is a root.
The one thing to do with the figure
Ask for a prediction before every press. The figure opens on base 64 with n = 3, so the labels 1, 4, 16 and 64 are already showing. Ask why each step multiplies by 4, then ask for the value at 4/3 before anyone sets m = 4. Then change n to 4 and ask why the step is not a whole number.
In the second figure, choose 1000 and m = 4 and ask which route they would rather do by hand.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: growth over five hours. Collect predictions of the hourly factor and of 323/5. |
| 12 min | Both figures with predictions; the key idea. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose C or D, return to the first figure. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | x⁸; 1/8; 12. |
|---|---|
| Hook | The hourly factor is the 5th root of 32, which is 2; 323/5 = 2³ = 8. |
| Worked examples | 64; 1/(9x⁴); x = 3/2. |
| Faded example | 27/8. |
| 1. Reading a fractional index | B. A multiplies by 2; C and D swap the root and the power. |
| 2. A fractional power with a numerator | 343. 7³. |
| 3. A negative fractional power | 0.008. 1/125. |
| 4. A power of a product | 25. (125x⁹)2/3 = 25x⁶. |
| 5. Adding fractional exponents | 5. 3/6 + 2/6. |
| 6. Roots in index form | C. A adds tops and bottoms; B subtracts; D multiplies the exponents. |
| 7. Solving an exponential equation | 2.5. 2x = 5. |
| 8. Undoing a fractional power | 9. 272/3. |
| 9. A reciprocal root as a power | D. A swaps root and power; B misses the reciprocal; C multiplies 3 and 4. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Treated the 2 as a multiplier, or swapped root and power. | Ask: which number is the root, and which is the power? Write it as a root symbol and an exponent. |
| 7 or 2 401 in question 2 | Took only the root, or used the denominator as the power. | Say “root first, then power” aloud while writing both. |
| −125 or 125 in question 3 | Made the answer negative, or ignored the minus sign. | Ask: what does a negative exponent do in 2−3? |
| 5 or 15 625 in question 4 | Did the root only, or the power only. | Do the root first, write the result, then apply the power. |
| 1 or 2 in question 5 | Multiplied the exponents, or added the tops. | Write 1/2 and 1/3 with a common denominator. |
| A, B or D in question 6 | Added tops and bottoms, subtracted, or multiplied. | Write each root as a power first, then add the exponents. |
| 8 or 5 in question 7 | Divided 32 by 4, or forgot that 4 = 2². | Write both sides as powers of 2. |
| 3 or 40.5 in question 8 | Undid only the 3, or multiplied by 3/2. | Ask: what power undoes a power of 3/2? |
| A, B or C in question 9 | Swapped root and power, forgot the reciprocal, or multiplied. | Write the cube root as a power first, then the reciprocal. |
Questions worth asking
- Why must x to the power 1/2 be the square root? Listen for “if the law (xa)b = xab is to work, (x1/2)² = x1”.
- Is 82/3 easier by root first or power first? Root first: the cube root of 8 is 2, and 2² = 4. Power first needs 8² = 64, then the cube root.
- What does x0 mean, in terms of steps? No steps, so the value is the starting 1.
- Why does the definition need x > 0? Otherwise the same exponent written two ways can give different answers.
Differentiation
A way in for anyone stuck
- Always write the root symbol and the power on separate lines before calculating.
- Keep a table of perfect powers (squares, cubes, 4th powers of 2 to 5) on the desk.
- Start with unit fractions, 1/n, before numerators other than 1.
Stretch
- Show that 21/2 × 81/2 = 4 using the laws of indices.
- Solve 32x − 12 × 3x + 27 = 0.
Students learning in English as an additional language
“Index” (plural “indices”) is another word for exponent or power. “Root” here means the number that multiplies itself to make another. “Reciprocal” means 1 divided by the number. Say 3/2 as “three halves” and x3/2 as “x to the power three halves”.
| Word | Plain meaning | Example |
|---|---|---|
| index | the small raised number that says how many times to multiply | the 5 in 2⁵ |
| indices | more than one index | the 3 and the 7 in 4³ and 4⁷ |
| root | the number that, multiplied by itself n times, gives another number | the cube root of 27 |
| reciprocal | 1 divided by a number | the reciprocal of 4 is 1/4 |
| base | the number that is being raised to a power | the 10 in 10⁻² |
| equivalent | different in form but equal in value | √x and x1/2 |
Sentence starters for explaining aloud or in writing:
- The denominator is …, so the root is …
- I can write … as a power of …, so …
- By the law of indices, …
Two things not to say
“Flip it and square root.” Say what each part of the fraction does: the denominator is the root and the numerator is the power.
“A negative power makes it negative.” It makes the reciprocal; the sign of the answer is unchanged.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- EvaluateEvaluate 361/2.
6. - Index formWrite √x as a power of x.
x1/2.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA culture of cells in Mumbai triples every 4 hours. Write the factor for 2 hours in index form and as a surd.
31/2 = √3, which is about 1.73. - Earlier work (Lesson 3)Expand (2 + √3)².
7 + 4√3.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningSimplify (16x8)−3/4.
16−3/4 = 1/8, and (x⁸)−3/4 = x−6, so 1/(8x⁶). - EquationSolve 8x = 1/4.
23x = 2−2, so x = −2/3.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider or button, then the arrow keys or Enter) and by touch. “Show me the steps” gives the same states in words.
The With a GDC section is closed by default and optional. Students without a GDC lose nothing; every answer is reached by hand.