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Grade 8 chapters and video availability01 · Read and understand
What you will learn
- Explain how to use cube roots to recover a signed input.
- Solve the two practice problems and explain how the assumptions affect the answers.
Before you start
Integer multiplication and powers.
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
Find the real solution of x³ = −125 and compare its sign behavior with a squared equation.
Why this math matters
Cubing preserves a real number's sign, so a negative real number has one negative real cube root. Distinguish odd and even powers before deciding how many real inverse values are possible.
Set up the model
A useful answer starts with clear assumptions:
- The requested solution is real.
- The exponent 3 means three repeated factors.
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.
Use cube roots to recover a signed input
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find the real solution of x³ = −125 and compare its sign behavior with a squared equation.
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
5³ = 125
First identify the positive magnitude that cubes to 125.
Apply the relationship
(−5)³ = (−5)(−5)(−5) = −125
Three negative factors leave a negative product.
Check and interpret
x = ∛(−125) = −5
Unlike an even power, a cube does not erase the sign of its real input.
The result
x = ∛(−125) = −5
Unlike an even power, a cube does not erase the sign of its real input.
Common mistakes to catch
- An odd root of a negative real number is real; the restriction for square roots does not apply.
- Volume uses cubic units, while its cube root is a length.
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
A cube has volume 216 cm³. Find its edge length.
Show a hint
Find the positive number whose cube is 216.
Reveal answer and explanation
6 cm
6³ = 216; a geometric edge length is positive.
Practice 2
Evaluate ∛(−8) + √16.
Show a hint
Use a negative cube root and a principal square root.
Reveal answer and explanation
2
∛(−8) = −2 and √16 = 4, so their sum is 2.
Take the idea with you
Distinguish odd and even powers before deciding how many real inverse values are possible.
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: Locate an irrational square root between decimals
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