Horton Ray
03/21/2023 · Senior High School
ALL integers have only cube root. \( > \) Why? What number(s) can you cube to get 8 ? What number(s) can you cube to get -8 ?
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Step-by-step Solution
Every integer has exactly one real cube root because the function \( f(x) = x^3 \) is a continuous and strictly increasing function. This means that for any real number \( y \), there is exactly one real number \( x \) such that \( x^3 = y \).
To answer your specific questions:
1. **What number(s) can you cube to get 8?**
The number that you can cube to get 8 is \( 2 \), because:
\[
2^3 = 2 \times 2 \times 2 = 8.
\]
2. **What number(s) can you cube to get -8?**
The number that you can cube to get -8 is \( -2 \), because:
\[
(-2)^3 = -2 \times -2 \times -2 = -8.
\]
In summary, the cube root of 8 is \( 2 \), and the cube root of -8 is \( -2 \). Each integer has a unique cube root, which can be either positive, negative, or zero.
Quick Answer
Every integer has a unique cube root. The cube root of 8 is 2, and the cube root of -8 is -2.
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