Question

Approximate Values of Irrational Numbers Practice Complete this assessment to review what you've learned. It will not count toward your grade. 1 of 51 of 5 Items Question Using a number line, what is the approximate number value of 103−−−√ ? Provide your answer in decimal form, rounded to the nearest hundredth.(1 point) 103−−−√ is approximately . Remaining Attempts : 3

Answer 1

To approximate the value of \( \sqrt{103} \), we can find two perfect squares that are close to 103. The perfect squares 100 (which is \( 10^2 \)) and 121 (which is \( 11^2 \)) surround 103.

1. The square root of 100 is 10.
2. The square root of 121 is 11.

Since \( \sqrt{103} \) falls between 10 and 11, we can start narrowing it down.

Next, let’s try squaring some numbers between 10 and 11 to see which might be closer to 103:

• \( 10.1^2 = 102.01 \)
• \( 10.2^2 = 104.04 \)

Since 102.01 is less than 103 and 104.04 is greater than 103, we know that \( \sqrt{103} \) is between 10.1 and 10.2.

Next, let’s try another midpoint:

• \( 10.15^2 = 103.0225 \) (a little too high)
• \( 10.14^2 = 102.8596 \) (a little too low)

From this, we can conclude that \( \sqrt{103} \) is approximately between 10.14 and 10.15.

For a more precise approximation, we can use a calculator to find the approximate value:

\[ \sqrt{103} \approx 10.1489 \]

Rounding to the nearest hundredth, we have:

\[ \sqrt{103} \approx 10.15 \]

So, the answer is \( \sqrt{103} \) is approximately 10.15.