Estimate the cube root of 15625.
The correct answer and explanation is:
Answer: The cube root of 15625 is approximately 25.
Explanation:
The cube root of a number xx is a number yy such that: y3=xy^3 = x
Here, we want to find 156253\sqrt[3]{15625}, a number yy such that: y3=15625y^3 = 15625
Step 1: Check for perfect cubes near 15625
To estimate the cube root, it’s helpful to recall some perfect cubes:
- 103=100010^3 = 1000
- 203=800020^3 = 8000
- 253=?25^3 = ?
- 303=2700030^3 = 27000
Since 15625 is between 8000 and 27000, the cube root will be between 20 and 30.
Step 2: Calculate 25325^3
Let’s compute 25325^3: 253=25×25×2525^3 = 25 \times 25 \times 25
First, 25×25=62525 \times 25 = 625
Then, 625×25=?625 \times 25 = ?
Calculate 625×25625 \times 25:
- 625×20=12500625 \times 20 = 12500
- 625×5=3125625 \times 5 = 3125
Add them: 12500+3125=1562512500 + 3125 = 15625
So, 253=1562525^3 = 15625
Step 3: Conclusion
Since 253=1562525^3 = 15625, the cube root of 15625 is exactly 25.
Additional context on cube roots and estimation:
- Cube roots can be estimated by identifying perfect cubes close to the given number.
- If the number is not a perfect cube, estimation involves interpolation between known cubes.
- For example, if you wanted 160003\sqrt[3]{16000}, you’d note 253=1562525^3 = 15625 and 263=1757626^3 = 17576, so the cube root of 16000 would be slightly above 25.
- Tools like prime factorization or trial multiplication are helpful in exact calculations.
- Knowing cubes of integers from 1 to 30 is useful for quick mental math estimations.
In summary, 156253=25\sqrt[3]{15625} = 25 exactly, as 15625 is a perfect cube.