Using rational approximations, what is an inequality statement comparing 81−−√
and 8–√8
? Enter <, >, or =.
explain how < is the correct answer
To compare the two expressions 81−−√ and 8–√8 using rational approximations, we can find the square root of both numbers and compare them.
First, let's find the square root of 81. The closest square root that is a rational number is 9, since 9 squared is 81.
Next, let's find the square root of 8. The closest square root that is a rational number is 2, since 2 squared is 4, which is less than 8.
Therefore, we have 81−−√ = 9 and 8–√8 = 2.
Since 9 is greater than 2, we can conclude that 81−−√ > 8–√8.
Thus, the correct inequality statement is 81−−√ > 8–√8.
To compare 81−−√ and 8–√8 using rational approximations, we can find the decimal approximations of each square root.
First, let's approximate 81−−√:
We know that 9 x 9 = 81, so 81−−√ is approximately 9.
Next, let's approximate 8–√8:
We can rewrite 8–√8 as 8–(2√2). Since 2 x 2 = 4, we can approximate 2√2 as 2 x √2 ≈ 2 x 1.4 ≈ 2.8. Therefore, 8–√8 is approximately 8 - 2.8 = 5.2.
Comparing the approximations, we have:
9 > 5.2
Therefore, the correct inequality statement is 81−−√ > 8–√8.
To compare the values of 81−−√ and 8–√8 using rational approximations, we can find the decimals that are close approximations for each of the values.
Let's start with 81−−√. To find a rational approximation, we need to find a value that is close to the square root of 81. We know that the square root of 81 is exactly 9, but we want a rational approximation. One such approximation is the integer 9 itself.
Now let's consider 8–√8. The square root of 8 is not an integer, so we need to find a rational approximation. One way to do this is to find two perfect squares that surround 8 on the number line. We know that 4^2 = 16 and 3^2 = 9. So, we can say that 3 < 8–√8 < 4.
Therefore, we have: 3 < 8–√8 < 4 < 9.
Since 3 < 4 < 9, we can conclude that 8–√8 is less than 81−−√. Therefore, the correct inequality statement is 8–√8 < 81−−√, which can be written as <.