write the power of $i$ as $i$, $-1$, $-i$, or $1$.

\i^{-37}\

$i^{-37} = \square$
(simplify your answer. type an exact answer, using radicals and $i$ as needed.)

Question

write the power of $i$ as $i$, $-1$, $-i$, or $1$.

\i^{-37}\

$i^{-37} = \square$
(simplify your answer. type an exact answer, using radicals and $i$ as needed.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Powers of i", "Complex Numbers" ], "new_concepts": [], "current_concepts": [ "Powers of i", "Complex Numbers" ] } ## Rewrite the negative exponent Using the **Complex Numbers** knowledge point $$ i^{-37} = \frac{1}{i^{37}} $$ ## Simplify the positive power of i Using the **Powers of i** knowledge point $$ \begin{aligned} 37 &= 4 \times 9 + 1 \ i^{37} &= (i^4)^9 \times i^1 = 1^9 \times i = i \end{aligned} $$ ## Evaluate the reciprocal Using the **Complex Numbers** knowledge point $$ \frac{1}{i} = \frac{1 \cdot i}{i \cdot i} = \frac{i}{i^2} = \frac{i}{-1} = -i $$ Write the power of $i$ as $i$, $-1$, $-i$, or $1$. $i^{-37} =$ $-i$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Powers of i" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Rewrite the negative exponent

Simplify the positive power of i

Evaluate the reciprocal

Answer: