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Grade 8 chapters and video availability01 · Read and understand
What you will learn
- Explain how to multiply and renormalize scientific notation.
- Solve the two practice problems and explain how the assumptions affect the answers.
Before you start
Common-base exponent rules and scientific notation.
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
A simulation uses 4 × 10³ samples with 6 × 10⁻⁵ seconds of modeled work per sample. Find the total modeled time.
Why this math matters
Products in scientific notation combine coefficients and powers separately, then restore the standard coefficient range. Separate coefficient arithmetic, exponent arithmetic, and unit cancellation when multiplying very large or small quantities.
Set up the model
A useful answer starts with clear assumptions:
- Each sample uses the same modeled time.
- The model excludes setup and communication time.
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.
Multiply and renormalize scientific notation
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A simulation uses 4 × 10³ samples with 6 × 10⁻⁵ seconds of modeled work per sample. Find the total modeled time.
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 quantities
time = (4 × 10³)(6 × 10⁻⁵) seconds
Sample units cancel against seconds per sample.
Apply the relationship
= 24 × 10⁻² seconds
Multiply 4 by 6 and add exponents 3 and −5.
Check and interpret
= 2.4 × 10⁻¹ seconds = 0.24 seconds
Move the coefficient into the normalized range while preserving the value.
The result
= 2.4 × 10⁻¹ seconds = 0.24 seconds
Move the coefficient into the normalized range while preserving the value.
Common mistakes to catch
- The exponent changes again if the coefficient needs normalization.
- Scientific notation does not remove the need to track measurement units.
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
Calculate (9 × 10⁷)/(3 × 10²).
Show a hint
Divide coefficients and subtract exponents.
Reveal answer and explanation
3 × 10⁵
9/3 = 3 and 7 − 2 = 5.
Practice 2
Calculate (7 × 10⁻³)(2 × 10⁴).
Show a hint
Multiply first, then normalize.
Reveal answer and explanation
1.4 × 10²
14 × 10¹ = 1.4 × 10² = 140.
Take the idea with you
Separate coefficient arithmetic, exponent arithmetic, and unit cancellation when multiplying very large or small quantities.
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: Distinguish a square root from solving a squared equation
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