Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 9 chapters and video availability01 · Read and understand
What you will learn
- Separate perfect-square factors from a nonnegative radicand.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Signed arithmetic, fraction and decimal operations, one-step equations, and introductory coordinate graphs.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Simplify √72 exactly.
Why this math matters
Preserve exact distances in geometric calculations until a numerical approximation is needed. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use the real-number system and the restrictions or data model stated in the question unless another domain is specified.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Extract square factors from radicals
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Simplify √72 exactly.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the conditions
72 = 36 × 2
Choose a perfect-square factor to extract.
Develop the calculation
√72 = √36 × √2
The product rule is valid here because both factors are nonnegative.
Check and interpret
√72 = 6√2
Two has no square factor greater than one, so the radical is fully simplified.
The result
√72 = 6√2
Two has no square factor greater than one, so the radical is fully simplified.
Common mistakes to catch
- Do not split a radical across addition.
- Exact radical form and decimal approximation are different representations.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Simplify √200.
Show a hint
Factor out one hundred.
Reveal answer and explanation
10√2
√(100×2)=10√2.
Practice 2
Is √(9+16) equal to √9+√16?
Show a hint
Calculate each side independently.
Reveal answer and explanation
No; 5 versus 7
The square-root product rule does not distribute over addition.
Take the idea with you
Preserve exact distances in geometric calculations until a numerical approximation is needed.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Prove a number is irrational
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