Squares and Square Roots: IM Alg2.3.6
[url=https://im.kendallhunt.com/HS/teachers/3/3/6/preparation.html]“Squares and Square Roots”[/url] from IM Algebra II © 2019 [url=https://www.illustrativemathematics.org/]Illustrative Mathematics[/url]. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0[/url] license.
Table of Contents
- Alg2.3.6 Lesson
- IM Alg2.3.6 Lesson: Squares and Square Roots
- Alg2.3.6 Practice
- IM Alg2.3.6 Practice: Squares and Square Roots
IM Alg2.3.6 Lesson: Squares and Square Roots
Find the solutions of each equation mentally.
[math]x^2=4[/math]
[math]x^2=2[/math]
[math]x^2=0[/math]
[math]x^2=-1[/math]
Clare was adding [math]\sqrt{4}[/math] and [math]\sqrt{9}[/math], and at first she wrote [math]\sqrt{4}+\sqrt{9}=2+3[/math]. 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:[br]2 + 3 = 5[br]2 + (-3) = -1[br](-2) + 3 = 1[br](-2) + (-3) = -5[br][br]Then she wondered, “Which of these are the same as [math]\sqrt{4}+\sqrt{9}[/math]? All of them? Or only some? Or just one?”[br][br]How would you answer Clare’s question? Give reasons that support your answer.
How many solutions are there to each equation?
[math]x^3=8[/math]
[math]y^3=\text{-}1[/math]
[math]z^4=16[/math]
[math]w^4=\text{-}81[/math]
[size=150]Write a rule to determine how many solutions there are to the equation [math]x^n=m[/math] where [math]n[/math] and [math]m[/math] are non-zero integers.[br][/size]
[size=150]The graph of [math]b=\sqrt{a}[/math] is shown.[/size]
Complete the table with the exact values and label the corresponding points on the graph with the exact values.
Label the point on the graph above that shows the solution to [math]\sqrt{a}=4[/math].
Label the point on the graph above that shows the solution to [math]\sqrt{a}=5[/math].[br]
Label the point on the graph above that shows the solution to [math]\sqrt{a}=\sqrt{5}[/math].[br]
The graph of is shown t=s².
Label the point(s) on the graph that show(s) the solution(s) to [math]s^2=25[/math].[br]
Label the point(s) on the graph that show(s) the solution(s) to [math]\sqrt{t}=5[/math].[br]
Label the point(s) on the graph that show(s) the solution(s) to [math]s^2=5[/math].
IM Alg2.3.6 Practice: Squares and Square Roots
[size=150]Select [b]all [/b]solutions to the equation [math]x^2=7[/math].[/size]
Find the solution(s) to each equation, if there are any.
[math]x^2=9[/math]
[math]\sqrt{x}=3[/math]
[math]\sqrt{x}=\text{-}3[/math]
[size=150]If [math]c[/math] is a positive number, how many solutions does [math]x^2=c[/math]have? Explain.[br][/size]
[size=150]If [math]c[/math] is a positive number, how many solutions does [math]\sqrt{x}=c[/math] have? Explain.[br][/size]
[size=150]Suppose that a friend missed class and never learned what [math]37^{\frac{1}{3}}[/math] means.[/size][br][br]Use exponent rules your friend would already know to calculate [math](37^{\frac{1}{3}})^3[/math].
Explain why this means that [math]37^{\frac{1}{3}}[/math] is the cube root of 37.[br]
Evaluate [math]8^{\frac{5}{3}}[/math].
Write each expression without using exponents.
[math]5^{\frac{2}{3}}[/math]
[math]4^{\text{-}\frac{3}{2}}[/math]