Question

enhanced algebra: concepts and connections - unit 3 georgias k-12 mathematics standards
name:
date:
formative assessment
solve and explain the reasoning behind the solutions. hint: some may have multiple solutions.
a. $a^2 = 6$
b. $b^3 = 7$
c. $c^3 = 8$
d. $d^2 = 7$
e. $e^3 = -8$
f. $f^2 = 9$
g. $g^3 = 11$

Explanation:

Part A: \( a^2 = 6 \)

Step1: Take square root of both sides

To solve for \( a \), we take the square root of both sides of the equation \( a^2 = 6 \). Remember that when taking the square root of a positive number, we get two solutions: one positive and one negative.
\[
a = \pm\sqrt{6}
\]

Step2: Simplify (if needed)

\( \sqrt{6} \) cannot be simplified further into a whole number or a simpler radical form, so the solutions are \( a = \sqrt{6} \) and \( a = -\sqrt{6} \).

Part B: \( b^3 = 7 \)

Step1: Take cube root of both sides

To solve for \( b \), we take the cube root of both sides of the equation \( b^3 = 7 \). Unlike square roots, cube roots of positive numbers have only one real solution (and a cube root of a negative number would have one real solution as well, which is negative).
\[
b = \sqrt[3]{7}
\]

Step2: Simplify (if needed)

\( \sqrt[3]{7} \) cannot be simplified further into a whole number or a simpler radical form, so the solution is \( b = \sqrt[3]{7} \).

Part C: \( c^3 = 8 \)

Step1: Take cube root of both sides

To solve for \( c \), we take the cube root of both sides of the equation \( c^3 = 8 \). We know that \( 2^3 = 8 \), so:
\[
c = \sqrt[3]{8} = 2
\]

Step2: Verify (optional but good practice)

Check if \( 2^3 = 8 \). \( 2 \times 2 \times 2 = 8 \), which is true. So the solution is \( c = 2 \).

Part D: \( d^2 = 7 \)

Answer:

Step1: Take cube root of both sides

To solve for \( g \), we take the cube root of both sides of the equation \( g^3 = 11 \).
\[
g = \sqrt[3]{11}
\]

Step2: Simplify (if needed)

\( \sqrt[3]{11} \) cannot be simplified further into a whole number or a simpler radical form, so the solution is \( g = \sqrt[3]{11} \).

Final Answers:

• A: \( a = \sqrt{6} \) or \( a = -\sqrt{6} \)
• B: \( b = \sqrt[3]{7} \)
• C: \( c = 2 \)
• D: \( d = \sqrt{7} \) or \( d = -\sqrt{7} \)
• E: \( e = -2 \)
• F: \( f = 3 \) or \( f = -3 \)
• G: \( g = \sqrt[3]{11} \)