Table of contents
- What Does It Mean to Find the Cube Root of 42.875?
- Find the Cube Root of 42.875: Quick Answer and Verification
- How to Find the Cube Root of 42.875 by Prime Factorization
- Find the Cube Root of 42.875 Using the Estimation Method
- How to Find the Cube Root of 42.875 with Halley’s Method
- How to Verify Your Answer
- Real-Life Example: Where Would You Use This?
- Common Mistakes When You Find the Cube Root of 42.875
- Related Cube Roots for Practice
- Conclusion
- Frequently Asked Questions
If you’re trying to find the cube root of 42.875, here’s the short answer: ∛42.875 = 3.5.
That’s an exact value, not a rounded one. Check it yourself: 3.5 × 3.5 × 3.5 = 42.875.
Decimals make cube roots look scarier than they are. Below I’ll show three ways to reach the answer, so you can use whichever suits your exam, homework or curiosity.
What Does It Mean to Find the Cube Root of 42.875?
A cube root asks one question: which number, multiplied by itself three times, gives me this number?
For 42.875, you want a number y where:
y × y × y = 42.875
You can write this three ways:
- With the radical sign: ∛42.875
- In exponent form: 42.875^(1/3)
- As an equation: y³ = 42.875
All three mean the same thing. If your textbook uses a different style, it’s the same idea.
Find the Cube Root of 42.875: Quick Answer and Verification
Expression | Value |
∛42.875 | 3.5 |
3.5³ | 42.875 |
Is 42.875 a perfect cube? | Yes (of the decimal 3.5, or of the fraction 7/2) |
Is the cube root rational? | Yes, since 3.5 = 7/2 |
How to Find the Cube Root of 42.875 by Prime Factorization
This is the most reliable method, and the one I’d recommend if your teacher wants “full working.”
Step 1: Remove the decimal.
42.875 has three digits after the decimal point, so write it over 1000:
42.875 = 42875 / 1000
Step 2: Factorize the numerator, 42875.
- 42875 ÷ 5 = 8575
- 8575 ÷ 5 = 1715
- 1715 ÷ 5 = 343
- 343 = 7 × 7 × 7
So 42875 = 5 × 5 × 5 × 7 × 7 × 7 = 5³ × 7³
Step 3: Group the factors into triplets.
5³ × 7³ = (5 × 7)³ = 35³
Step 4: Factorize the denominator.
1000 = 10 × 10 × 10 = 10³
Step 5: Take the cube root of each part.
∛(42875 / 1000) = ∛(35³) / ∛(10³) = 35 / 10 = 3.5
Answer: ∛42.875 = 3.5
Both the numerator and denominator are perfect cubes, so the decimal cube root comes out clean.
Find the Cube Root of 42.875 Using the Estimation Method
This method is quick, and it works well when you’re short on time in an exam.
Step 1: Find the two perfect cubes on either side of the number.
- 3³ = 27
- 4³ = 64
Since 42.875 is between 27 and 64, the cube root is between 3 and 4.
Step 2: Narrow it down.
42.875 is a bit past halfway between 27 and 64, so try a value near the middle of 3 and 4.
Step 3: Test 3.5.
- 3.5 × 3.5 = 12.25
- 12.25 × 3.5 = 42.875 ✓
The cube root is 3.5.
A quick shortcut: the last digit of 42875 is 5, and only a number ending in 5 can have a cube ending in 5. That’s a good hint that the cube root ends in 5, which points you to 3.5.
How to Find the Cube Root of 42.875 with Halley’s Method
Halley’s method is a numerical technique for approximating cube roots. It’s usually reserved for numbers that are not perfect cubes, but it works well here and shows how quickly it converges.
The formula is:
∛a ≈ x × (x³ + 2a) / (2x³ + a)
where a = 42.875 and x is a starting guess.
Round 1: start with x = 3
∛42.875 ≈ 3 × (27 + 85.75) / (54 + 42.875)
= 3 × 112.75 / 96.875
≈ 3.4916
Round 2: use x = 3.4916
- x³ ≈ 42.567
- ∛42.875 ≈ 3.4916 × (42.567 + 85.75) / (85.134 + 42.875)
- ≈ 3.4916 × 128.317 / 130.009 …
Rather than grinding through more decimals, notice that the result lands on 3.5000 after a second round, which confirms the exact answer from Method 1.
One round of Halley’s method gets you to about 3.49, and a second round gets you to 3.5. That’s why it’s so popular in calculators and software.
How to Verify Your Answer
Never skip the check. It takes ten seconds:
3.5 × 3.5 = 12.25
12.25 × 3.5 = 42.875 ✓
If cubing your answer returns the original number, you’re done.
Real-Life Example: Where Would You Use This?
Suppose a cube-shaped storage box has a volume of 42.875 cubic centimetres. How long is each side?
Side length = ∛42.875 = 3.5 cm
The same idea applies in packaging design, aquarium sizing, 3D modelling and any physics problem where you’re given a volume and need a length.
Common Mistakes When You Find the Cube Root of 42.875
I see these mistakes again and again, so watch out for them.
- Assuming it can’t be exact because there’s a decimal.
A decimal can be a perfect cube. 42.875 is a perfect cube because 3.5³ lands on it exactly. - Dividing by 3.
Cube root is not “divide by 3.” 42.875 ÷ 3 is about 14.29, which is nowhere near the answer. - Confusing cube root with square root.
√42.875 is about 6.55. That’s a different question entirely. Always check the little 3 on the radical sign. - Miscounting decimal places when converting to a fraction.
42.875 has three decimal places, so the denominator is 1000, not 100. Because 1000 = 10³, you can take the cube root of the denominator easily. 100 isn’t a perfect cube, which would make the problem much harder. - Rounding too soon.
If you round halfway through (say, to 3.5 using a guess instead of a check), you may be right by luck but lose marks for missing work. Verify by cubing.
Related Cube Roots for Practice
Once you’ve nailed this one, try these similar decimal perfect cubes:
Number | Cube Root |
0.125 | 0.5 |
3.375 | 1.5 |
15.625 | 2.5 |
42.875 | 3.5 |
91.125 | 4.5 |
Spot the pattern? They all follow the form (n + 0.5)³. If you remember this, many “scary” decimal cube roots become easy to recognise.
Conclusion
To find the cube root of 42.875, convert it to 42875/1000, factorize the numerator into 5³ × 7³, and read off 35/10 = 3.5. Or estimate between 3³ and 4³, test 3.5, and confirm by cubing.
Next time you meet a decimal cube root, try the fraction trick first. It turns the problem into plain whole-number factorization.
Frequently Asked Questions
It is exactly 3.5.
Yes. 3.5 × 3.5 × 3.5 = 42.875, so it’s a perfect cube of the rational number 3.5 (or 7/2).
Rational. It can be written as 7/2, a ratio of two integers.
42.875^(1/3) = 3.5.
Convert it to 42875/1000, factorize 42875 into 5³ × 7³, then take the cube root of the top and bottom to get 35/10 = 3.5.
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