Lesson 6
Squares and Square Roots
 Let’s compare equations with squares and square roots.
6.1: Math Talk: Four Squares
Find the solutions of each equation mentally.
\(x^2 = 4\)
\(x^2 = 2\)
\(x^2 = 0\)
\(x^2 = \text{}1\)
6.2: Finding Square Roots
Clare was adding \(\sqrt{4}\) and \(\sqrt{9}\), and at first she wrote \(\sqrt{4} + \sqrt{9} = 2+3\). But then she remembered that 2 and 2 both square to make 4, and that 3 and 3 both square to make 9. She wrote down all the possible combinations:
2 + 3 = 5
2 + (3) = 1
(2) + 3 = 1
(2) + (3) = 5
Then she wondered, “Which of these are the same as \(\sqrt{4} + \sqrt{9}\)? All of them? Or only some? Or just one?”
How would you answer Clare’s question? Give reasons that support your answer.
 How many solutions are there to each equation?
 \(x^3=8\)
 \(y^3=\text 1\)
 \(z^4 = 16\)
 \(w^4 = \text 81\)
 Write a rule to determine how many solutions there are to the equation \(x^n=m\) where \(n\) and \(m\) are nonzero integers.
6.3: One Solution or Two?

The graph of \(b=\sqrt{a}\) is shown.
 Complete the table with the exact values and label the corresponding points on the graph with the exact values.
\(a\) 1 4 9 12 16 20 \(\sqrt{a}\)  Label the point on the graph that shows the solution to \(\sqrt{a} = 4\).
 Label the point on the graph that shows the solution to \(\sqrt{a} = 5\).
 Label the point on the graph that shows the solution to \(\sqrt{a} = \sqrt{5}\).
 Complete the table with the exact values and label the corresponding points on the graph with the exact values.

The graph of \(t = s^2\) is shown.
 Label the point(s) on the graph that show(s) the solution(s) to \(s^2 = 25\).
 Label the point(s) on the graph that show(s) the solution(s) to \(\sqrt{t} = 5\).
 Label the point(s) on the graph that show(s) the solution(s) to \(s^2 = 5\).
Summary
The symbol \(\sqrt{11}\) represents the positive square root of 11. If we want to represent the negative square root, we write \(\text{} \sqrt{11}\).
The equation \(x^2 = 11\) has two solutions, because \(\sqrt{11}^2=11\), and also\(\left(\text{}\sqrt{11}\right)^2=11\).
The equation \(\sqrt{x} = 11\) only has one solution, namely 121.
The equation \(\sqrt{x} = \sqrt{11}\) only has one solution, namely 11.
The equation \(\sqrt{x} = \text{}11\) doesn’t have any solutions, because the left side is positive and the right side is negative, which is impossible, because a positive number cannot equal a negative number.