Index laws extend consistently beyond positive whole-number powers. A zero index gives 1 for a non-zero base, a negative index represents a reciprocal, and a fractional index represents a root.
Example
16^(3/4) = (fourth root of 16)^3 = 2^3 = 8, while 2^-3 = 1/8.
Key terms
Zero index:
A non-zero base raised to zero equals 1.
Negative index:
An index representing the reciprocal of a positive power.
Fractional index:
An index representing a root and a power.
Questions
1. What is the central idea in zero, negative and fractional indices?
how extended index laws express reciprocals and roots consistently
Guess from the size of the numbers.
Apply an unrelated rule without checking conditions.
Write only a final answer with no reasoning or units.
2. Which term means "A non-zero base raised to zero equals 1."?
Zero index
Negative index
Fractional index
Context
3. Which term means "An index representing the reciprocal of a positive power."?
Negative index
Zero index
Fractional index
Evidence
4. Which term means "An index representing a root and a power."?
Fractional index
Zero index
Negative index
Reflection
5. Which task best practises zero, negative and fractional indices?
Simplify a mixed set of zero, negative and fractional index expressions and verify using ordinary fractions or roots.
Guess from the size of the numbers.
Apply an unrelated rule without checking conditions.
Write only a final answer with no reasoning or units.
6. Which approach best supports learning in Maths?
Identify the relationships, choose a valid method, show each step and check the result in context.
Guess from the size of the numbers.
Apply an unrelated rule without checking conditions.
Write only a final answer with no reasoning or units.
7. Why is a worked example useful?
It makes the reasoning and deliberate choices visible.
It removes the need to think.
It guarantees every new problem is identical.
It replaces practice completely.
8. Which response applies how extended index laws express reciprocals and roots consistently?
Simplify a mixed set of zero, negative and fractional index expressions and verify using ordinary fractions or roots.
Guess from the size of the numbers.
Apply an unrelated rule without checking conditions.
Write only a final answer with no reasoning or units.
9. What makes guided practice useful?
It gives support while the learner tries the thinking for themselves.
It supplies answers before any attempt.
It avoids feedback and reflection.
It makes the final check unrelated.
10. How should the key terms support zero, negative and fractional indices?
They should make the explanation more precise and connected to evidence.
They should be listed without meaning.
They should replace examples.
They should be used only for spelling.
11. What is the best response when a first attempt is incomplete?
Use feedback or evidence to revise the reasoning.
Hide the attempt.
Repeat it without checking.
Choose an unrelated answer.
12. Which explanation is strongest?
A clear idea supported by a relevant example and reasoning.
A claim with no support.
A copied definition only.
A long response that avoids the question.
13. Why transfer the skill to a new example?
It shows whether the understanding can be used beyond the worked model.
It proves all examples are identical.
It makes the original lesson unnecessary.
It prevents reflection.
14. What should a checkpoint reveal?
Whether the learner is ready for the final check or needs another explanation.
Only whether the learner worked quickly.
Whether the topic title was memorised.
Nothing about understanding.
15. What makes a conclusion responsible?
It matches the evidence and acknowledges important limits.
It claims more than the evidence shows.
It ignores alternatives.
It is decided before the task.
16. How can zero, negative and fractional indices support independent learning?
It gives a repeatable way to interpret, create, solve or evaluate a new situation.
It works only for the example already shown.
It removes the need for judgement.
It depends on guessing.
17. What should happen when evidence challenges the first interpretation or method?
Review the reasoning and revise it when the evidence warrants change.
Discard the evidence automatically.
Keep the first answer regardless.
Stop checking the work.
18. Which reflection leads to useful improvement?
Identify a successful choice, evidence of its effect and one specific next step.
State only that the task was easy or hard.
List the title again.
Avoid referring to the work.
19. What distinguishes strong Year 9 Maths work?
Accurate knowledge, deliberate choices, evidence and clear reasoning.
Length without relevance.
Confidence without checking.
Memorisation without application.
20. Why should an application task remain manageable but substantial?
It should provide enough challenge to demonstrate real learning without creating unnecessary overload.
It should remove all challenge.
It should be long regardless of purpose.
It should repeat the quiz word for word.
21. What is the strongest outcome from zero, negative and fractional indices?
Use how extended index laws express reciprocals and roots consistently accurately in a purposeful new context.
Guess from the size of the numbers.
Apply an unrelated rule without checking conditions.
Write only a final answer with no reasoning or units.
Answer key (parent copy)
1. how extended index laws express reciprocals and roots consistently
2. Zero index
3. Negative index
4. Fractional index
5. Simplify a mixed set of zero, negative and fractional index expressions and verify using ordinary fractions or roots.
6. Identify the relationships, choose a valid method, show each step and check the result in context.
7. It makes the reasoning and deliberate choices visible.
8. Simplify a mixed set of zero, negative and fractional index expressions and verify using ordinary fractions or roots.
9. It gives support while the learner tries the thinking for themselves.
10. They should make the explanation more precise and connected to evidence.
11. Use feedback or evidence to revise the reasoning.
12. A clear idea supported by a relevant example and reasoning.
13. It shows whether the understanding can be used beyond the worked model.
14. Whether the learner is ready for the final check or needs another explanation.
15. It matches the evidence and acknowledges important limits.
16. It gives a repeatable way to interpret, create, solve or evaluate a new situation.
17. Review the reasoning and revise it when the evidence warrants change.
18. Identify a successful choice, evidence of its effect and one specific next step.
19. Accurate knowledge, deliberate choices, evidence and clear reasoning.
20. It should provide enough challenge to demonstrate real learning without creating unnecessary overload.
21. Use how extended index laws express reciprocals and roots consistently accurately in a purposeful new context.