Find all real square roots of 64
The Correct answer and Explanation is:
The square roots of a number (x) are the values that, when multiplied by themselves, yield (x). For the number 64, we can find the real square roots by solving the equation:
[
x^2 = 64
]
To determine the square roots, we can rewrite the equation:
[
x = \sqrt{64}
]
Calculating the principal square root of 64, we find:
[
\sqrt{64} = 8
]
This means that 8 is one square root of 64. However, it’s important to recognize that every positive number actually has two real square roots: one positive and one negative. Therefore, the other square root is:
[
x = -\sqrt{64} = -8
]
In conclusion, the two real square roots of 64 are (8) and (-8).
Explanation
- Understanding Square Roots: The square root function asks for the number that, when multiplied by itself, produces the original number. For example, since (8 \times 8 = 64), 8 is a square root of 64. Similarly, because (-8 \times -8 = 64) (the product of two negative numbers is positive), -8 is also a square root.
- Mathematical Notation: The square root symbol (√) typically refers to the principal (or non-negative) square root. Hence, (\sqrt{64} = 8) gives us the positive root. However, the concept of square roots encompasses both the positive and negative roots.
- Verification: To confirm the findings, we can check the products:
- (8^2 = 64)
- ((-8)^2 = 64)
Both calculations affirm that both 8 and -8 are correct square roots.
- Context of Square Roots: Square roots play an essential role in various mathematical applications, including solving quadratic equations, geometry (e.g., calculating areas), and real-world problems involving distances and areas.
In summary, the real square roots of 64 are (8) and (-8), and both are essential for understanding the broader implications of square roots in mathematics.