Question

How does finding the square root of a number compare to finding the cube root of a number, using 64 as an example?

Answer 1

Finding the square root and the cube root of a number involves determining what value, when multiplied by itself a specified number of times, equals the original number. Let’s use 64 as our example to illustrate the difference between the two operations.

1. **Square Root**:
- The square root of a number \( x \) is a value \( y \) such that \( y^2 = x \).
- For 64, we identify which number squared gives us 64:
\[
y^2 = 64
\]
The solution is:
\[
y = 8 \quad (\text{since } 8 \times 8 = 64)
\]
- Therefore, the square root of 64 is \( 8 \), which can be expressed as:
\[
\sqrt{64} = 8
\]

2. **Cube Root**:
- The cube root of a number \( x \) is a value \( z \) such that \( z^3 = x \).
- For 64, we find which number cubed gives us 64:
\[
z^3 = 64
\]
The solution is:
\[
z = 4 \quad (\text{since } 4 \times 4 \times 4 = 64)
\]
- Thus, the cube root of 64 is \( 4 \), which can be expressed as:
\[
\sqrt[3]{64} = 4
\]

### Summary:
- The square root of 64 is \( 8 \) because \( 8 \times 8 = 64 \).
- The cube root of 64 is \( 4 \) because \( 4 \times 4 \times 4 = 64 \).

### Comparison:
- **Degree of Operation**: The square root involves squaring (degree 2), while the cube root involves cubing (degree 3).
- **Results**: The square root is typically larger than the cube root for numbers greater than one (in this case, \( 8 > 4 \)) since they involve different multiplications of the base number.
- **Usage**: Square roots are used in contexts where two factors equal a number, while cube roots relate to three factors.