drag each expression to the correct location on the table.

determine which expressions represent purely real numbers and which expressions represent non-real complex numbers.

purely real number | non-real complex number

$7 - 5i$ $\sqrt{-6}$ $0 + 9i$ $\sqrt{(-5)^2}$ $-i^2 + i^3$ $2 - 7i^2$ $-12$ $i^6$

Question

drag each expression to the correct location on the table.

determine which expressions represent purely real numbers and which expressions represent non-real complex numbers.

purely real number | non-real complex number

$7 - 5i$ $\sqrt{-6}$ $0 + 9i$ $\sqrt{(-5)^2}$ $-i^2 + i^3$ $2 - 7i^2$ $-12$ $i^6$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Complex Number Arithmetic" ], "new_concepts": [ "Real and Imaginary Numbers", "Powers of i" ], "current_concepts": [ "Complex Number Arithmetic", "Real and Imaginary Numbers", "Powers of i" ] } ## Evaluate each expression using powers of $i$ and square roots $$ \begin{aligned} &7 - 5i \quad \text{has a non-zero imaginary part } -5i \ &\sqrt{-6} = i\sqrt{6} \quad \text{has a non-zero imaginary part} \ &0 + 9i = 9i \quad \text{has a non-zero imaginary part} \ &\sqrt{(-5)^2} = \sqrt{25} = 5 \quad \text{is purely real} \ &-i^2 + i^3 = -(-1) + (-i) = 1 - i \quad \text{has a non-zero imaginary part} \ &2 - 7i^2 = 2 - 7(-1) = 2 + 7 = 9 \quad \text{is purely real} \ &-12 \quad \text{is purely real} \ &i^6 = (i^2)^3 = (-1)^3 = -1 \quad \text{is purely real} \end{aligned} $$ ## Classify expressions into categories $$ \begin{aligned} &\text{Purely Real Numbers: } \sqrt{(-5)^2},\ 2 - 7i^2,\ -12,\ i^6 \ &\text{Non-real Complex Numbers: } 7 - 5i,\ \sqrt{-6},\ 0 + 9i,\ -i^2 + i^3 \end{aligned} $$ | Purely Real Number | Non-real Complex Number | |---|---| | $\sqrt{(-5)^2}$ | $7 - 5i$ | | $2 - 7i^2$ | $\sqrt{-6}$ | | $-12$ | $0 + 9i$ | | $i^6$ | $-i^2 + i^3$ | { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Complex Numbers" ] }

Purely Real Number	Non-real Complex Number
\(2 - 7i^2\)	\(\sqrt{-6}\)
\(-12\)	\(0 + 9i\)
\(i^6\)	\(-i^2 + i^3\)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Evaluate each expression using powers of \(i\) and square roots

Classify expressions into categories

Answer: