Analyze Relationships To find \(\sqrt{15}\) , Beau found \(3^ 2 = 9 \)  and  \(4^ 2 = 16\). He said that since 15 is between 9 and 16, \(\sqrt{15}\)  must be between 3 and 4. He thinks a good estimate for \(\sqrt{15}\) is \(\large\frac{ 3 + 4}{2}= 3.5\). Is Beau’s estimate high, low, or correct? Explain.

Justify Reasoning What is a good estimate for the solution to the equation \(x^ 3 = 95\)?  How did you come up with your estimate?

The volume of a sphere is \(36 \pi \text{ ft}^ 3 \). What is the radius of the sphere? Use the formula \(V = \frac{4}{3}\pi r ^3\) to find your answer.

Draw Conclusions Can you find the cube root of a negative number? If so, is it positive or negative? Explain your reasoning.

Make a Conjecture Evaluate and compare the following expressions.
\(\sqrt{\frac{4}{25}}\)  and \(\frac{\sqrt{4}}{\sqrt{25}}\)     \(\sqrt{\frac{16}{81}}\)  and \(\frac{\sqrt{16}}{\sqrt{81}}\)     \(\sqrt{\frac{36}{49}}\)  and \(\frac{\sqrt{36}}{\sqrt{49}}\)      
Use your results to make a conjecture about a division rule for square roots. Since division is multiplication by the reciprocal, make a conjecture about a multiplication rule for square roots.

Persevere in Problem Solving

The difference between the solutions to the equation \(x^2=a\) is 30. What is \(a\)? Show that your answer is correct.