Select students to share a few estimates and exact solutions to the equations and to explain how they used the graph to help estimate. It is important to discuss some of the differences between cube roots and square roots. Here are some questions for discussion:

• “Think back to when we were looking at the graph of . From that graph, what do the solutions for look like if a is positive, zero, or negative?” (For positive values of , we notice that a horizontal line at crosses the graph of twice, so the equation has two real solutions, the positive square root and the negative square root of . For , we notice that intersects the graph of exactly once, so there is only one real solution to the equation , which is 0. When a is negative, there are no real solutions to the equation , because any horizontal line below the -axis doesn’t intersect the graph of .)
• “What about the real solutions to ? From the graph of , what do the solutions look like for different values of ?” (Horizontal lines always cross the graph of at one point, so the equation  always has exactly one real solution.)
• “When we saw the symbol earlier, we had to think about what it might mean. What do we think should mean?” (A positive number has two square roots on the number line, so we had to figure out what we wanted the square root symbol to mean. For any number we can find exactly one cube root on the number line, so it makes sense to make this what the cube root symbol means.)

Discuss the difference between cubing each side of to get and squaring each side of to get . Ask students, “Why is one valid while the other isn’t?” (If has a value of -4, then it cubes to make -64, and -64 has only one real cube root. In contrast, the equation has no real solutions because it’s impossible for the positive square root to be equal to a negative number when is a real value.)

If students have trouble starting on the more difficult equations, consider asking:

• “How did you choose the equations you thought would be the least and most difficult to solve?”

• “What number could you use instead of the expression with the root to make the equation true?”

The purpose of the discussion is to articulate how to solve equations involving a cube root of a variable.

Direct students’ attention to the reference created using Collect and Display. Ask students to share an equation they selected and how they solved it. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

Here are some questions for discussion:

• “How is the process of solving these equations the same as solving equations that involve square roots? How is it different?” (Solving these equations is different because they require cubing rather than squaring. It is also different because cube roots aren’t always positive, so the extra step to check that a positive number isn’t set equal to a negative number isn’t necessary. Solving is the same because it requires isolating the radical expression and then using exponents to make a new equation without radicals.)
• “Describe any difficulties you experienced and how you resolved them.” (I cubed a negative number and forgot that the result would be negative; when I plugged in my answer to check it, I saw that I must have done something wrong, so I went back and fixed it. I wasn’t sure how to get started on one of my difficult equations, but I remembered that I should try to isolate the cube root, and then I could see what to do.)
• “Describe a general method for solving equations like these that involve a variable inside of a cube root.” (First, isolate the root by appropriately adding, subtracting, multiplying, or dividing any constants outside of the root on each side of the equation. Next, cube each side of the equation. Then solve for the variable.)