Using rational approximations, what is an inequality statement comparing 81−−√ and 8–√8? Enter <, >, or =.
To compare these two numbers using rational approximations, we can find the decimal representation of each number and compare them.
First, let's find a rational approximation for √81. We know that the square root of 81 is 9, so √81 = 9.
Now let's find a rational approximation for √8. We know that √9 is 3, so √8 must be slightly less than 3. Approximating √8 to be 2.83.
Comparing 9 and 2.83, we can see that 9 > 2.83.
Therefore, the inequality statement comparing 81−−√ and 8–√8 is 81−−√ > 8–√8.
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To compare 81−−√ and 8–√8 using rational approximations, we need to find their decimal approximations.
First, let's find the decimal approximation for 81−−√:
Step 1: Start with an initial guess, let's say x = 9.
Step 2: Use the formula for Newton's method: x = (x + (81/x)) / 2.
For the first iteration: x = (9 + (81/9)) / 2 = (9 + 9) / 2 = 9.
For the second iteration: x = (9 + (81/9)) / 2 = (9 + 9) / 2 = 9.
Since the value of x does not change, we can conclude that 81−−√ = 9, with a good approximation.
Now, let's find the decimal approximation for 8–√8:
Step 1: Start with an initial guess, let's say x = 2.
Step 2: Use the formula for Newton's method: x = (x + (8/x)) / 2.
For the first iteration: x = (2 + (8/2)) / 2 = (2 + 4) / 2 = 3.
For the second iteration: x = (3 + (8/3)) / 2 = (3 + 2.6667) / 2 = 2.8333.
For the third iteration: x = (2.8333 + (8/2.8333)) / 2 = (2.8333 + 2.8259) / 2 = 2.8296.
For the fourth iteration: x = (2.8296 + (8/2.8296)) / 2 = (2.8296 + 2.8284) / 2 = 2.8295.
Therefore, using rational approximations, we can say that 81−−√ > 8–√8.
To compare the irrational numbers 81−−√ and 8–√8 using rational approximations, we can find decimal approximations for each number and compare them.
First, let's find the decimal approximation for 81−−√. We can calculate the square root using a calculator or use the method of successive approximation. Starting with a guess of 9, we can improve the approximation by iterating through the following steps:
Guess: 9
1st iteration: (9 + 81 / 9) / 2 = 9.5
2nd iteration: (9.5 + 81 / 9.5) / 2 ≈ 9.024
3rd iteration: (9.024 + 81 / 9.024) / 2 ≈ 9.005
4th iteration: (9.005 + 81 / 9.005) / 2 ≈ 9.00003
Therefore, the rational approximation for 81−−√ is approximately 9.00003.
Next, let's find the decimal approximation for 8–√8. Similarly, we use the method of successive approximation or a calculator. Starting with a guess of 2, we can iterate through the following steps:
Guess: 2
1st iteration: (2 + 8 / 2) / 2 = 2.5
2nd iteration: (2.5 + 8 / 2.5) / 2 ≈ 2.463
3rd iteration: (2.463 + 8 / 2.463) / 2 ≈ 2.461
4th iteration: (2.461 + 8 / 2.461) / 2 ≈ 2.461
Therefore, the rational approximation for 8–√8 is approximately 2.461.
Comparing these decimal approximations, we have the following inequality statement:
9.00003 > 2.461
So the inequality statement comparing 81−−√ and 8–√8 is 81−−√ > 8–√8.