Which two numbers does the square root of 128 lie between on a number line
The Correct Answer and Explanation is :
The correct answer is: 11.312
To determine which two numbers the square root of 128 lies between, we can start by estimating the square root.
- Calculating Square Root:
[
\sqrt{128}
] - Finding Perfect Squares:
We need to find two perfect squares that are closest to 128:
- (11^2 = 121)
- (12^2 = 144) Since (121 < 128 < 144), it follows that:
[
11 < \sqrt{128} < 12
]
- Estimating More Accurately:
To get a more accurate estimate of (\sqrt{128}), we can calculate it:
[
\sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \cdot \sqrt{2} = 8 \cdot \sqrt{2}
]
The approximate value of (\sqrt{2}) is about (1.414), thus:
[
8 \cdot 1.414 \approx 11.312
]
This calculation confirms that (\sqrt{128} \approx 11.312).
- Conclusion:
The square root of 128 lies between 11 and 12. More specifically, it is closer to 11.3, confirming that:
[
11 < \sqrt{128} < 12
]
Explanation:
Understanding square roots is fundamental in mathematics, particularly in geometry and algebra. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, determining the square root of 128 helps us locate its value on a number line.
When estimating square roots, identifying perfect squares allows us to create bounds. Perfect squares are integers that result from squaring whole numbers. Since (11^2) and (12^2) bracket 128, we can confidently state that the square root of 128 falls between these two integers.
Calculating the square root more precisely reveals that (\sqrt{128}) is approximately 11.312. This approximation helps visualize where the value lies on the number line, further reinforcing the conclusion that the square root of 128 is indeed between 11 and 12.