What are the positive and negative square roots of 64/25
The Correct Answer and Explanation is:
The square roots of a number xxx are the values that, when multiplied by themselves, give xxx. The square roots of a number always have two values: one positive and one negative.
For the number 6425\frac{64}{25}2564, we can find the square roots by first taking the square root of the numerator (64) and the denominator (25) separately, and then combining the results. 6425=6425=85\sqrt{\frac{64}{25}} = \frac{\sqrt{64}}{\sqrt{25}} = \frac{8}{5}2564=2564=58
The square root of 646464 is 8, and the square root of 252525 is 5. Therefore, the principal (positive) square root of 6425\frac{64}{25}2564 is 85\frac{8}{5}58.
However, for any positive number xxx, there are always two square roots: one positive and one negative. Therefore, the square roots of 6425\frac{64}{25}2564 are: ±85\pm \frac{8}{5}±58
This means that both 85\frac{8}{5}58 and −85-\frac{8}{5}−58 are square roots of 6425\frac{64}{25}2564.
Explanation:
The square root function has two outputs for any positive input because both positive and negative numbers, when squared, give the same result. For example:
- 82=648^2 = 6482=64
- (−8)2=64(-8)^2 = 64(−8)2=64
Thus, both +85+\frac{8}{5}+58 and −85-\frac{8}{5}−58 are valid solutions. The “positive” square root typically refers to the non-negative root, while the “negative” square root refers to the negative counterpart.
To summarize:
- Positive square root: 85\frac{8}{5}58
- Negative square root: −85-\frac{8}{5}−58
