Using rational approximations, what is an inequality statement comparing 21−−√
and 17−−√
? Enter <, >, or =.
21−−√ > 17−−√
Using rational approximations, what is an inequality statement comparing 21−−√ and 17−−√? Enter <, >, or =.
To compare the numbers 21−−√ and 17−−√ using rational approximations, we can use the fact that the square root of any non-perfect square is an irrational number and cannot be expressed as a fraction.
Let's find rational approximations for both numbers:
For 21−−√, we can start by estimating its value. Since the square root of 16 is 4, and 21 is slightly larger than 16, we know that 21−−√ must be slightly larger than 4. So, we can choose a rational approximation of 4.
For 17−−√, we can do a similar estimation. Since the square root of 16 is 4, and 17 is slightly larger than 16, we know that 17−−√ must be slightly larger than 4. So, we can also choose a rational approximation of 4 for 17−−√.
Therefore, based on our rational approximations, we can say that 21−−√ is greater than 17−−√. In other words, 21−−√ > 17−−√.
To compare the square roots, we need to find rational approximations for 21 and 17.
Let's start by finding the rational approximation for √21:
1. List the perfect square numbers greater than 21: 1, 4, 9, 16, 25.
2. Identify the two perfect squares that √21 lies between: √16 = 4 and √25 = 5.
3. Divide the range between those two perfect squares into smaller increments: √16, √17, √18, √19, √20, √21, √22, √23, √24, √25.
4. Approximate the value of √21 by comparing it to the incremental values: √21 is closer to √20 since it's between √20 and √21.
Therefore, we can approximate √21 as ≈ √20 ≈ 4.472.
Now let's find the rational approximation for √17:
1. List the perfect square numbers greater than 17: 1, 4, 9, 16, 25.
2. Identify the two perfect squares that √17 lies between: √16 = 4 and √25 = 5.
3. Divide the range between those two perfect squares into smaller increments: √16, √17, √18, √19, √20, √21, √22, √23, √24, √25.
4. Approximate the value of √17 by comparing it to the incremental values: √17 is closer to √16 since it's between √16 and √17.
Therefore, we can approximate √17 as ≈ √16 ≈ 4.
Comparing the rational approximations, we have 21−−√ ≈ 4.472 and 17−−√ ≈ 4.
Since 4.472 > 4, the inequality statement is:
21−−√ > 17−−√