Question

Which two consecutive integers does − sqrt 75
lie between?

Answer 1

To find two consecutive integers that \(-\sqrt{75}\) lies between, we first calculate \(\sqrt{75}\).

We can simplify \(\sqrt{75}\):

\[
\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}
\]

Next, we need to estimate \(\sqrt{3}\). We know that:

\[
1.7^2 = 2.89 \quad \text{and} \quad 1.8^2 = 3.24
\]

Thus, we can determine that \(1.7 < \sqrt{3} < 1.8\). To get a more precise value, we can estimate \(\sqrt{3}\) as approximately \(1.732\).

Now, substituting that value back:

\[
\sqrt{75} \approx 5 \times 1.732 = 8.66
\]

Therefore:

\[
-\sqrt{75} \approx -8.66
\]

Now, we need to find the two consecutive integers that \(-8.66\) lies between. The integers that surround \(-8.66\) are \(-9\) and \(-8\).

Thus, \(-\sqrt{75}\) lies between the integers \(-9\) and \(-8\).

In conclusion, \(-\sqrt{75}\) lies between the consecutive integers \(-9\) and \(-8\).