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

\i^3 \cdot i^{19}\

\i^3 \cdot i^{19} = \boxed{\quad}\

Question

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

\i^3 \cdot i^{19}\

\i^3 \cdot i^{19} = \boxed{\quad}\

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" ], "new_concepts": [], "current_concepts": [ "Imaginary Unit", "Powers of i", "Complex Numbers" ] } ## Simplify the product of powers Using the *Powers of i* knowledge point $$ i^3 \cdot i^{19} = i^{3 + 19} = i^{22} $$ ## Reduce the exponent modulo 4 Using the *Powers of i* knowledge point $$ \begin{aligned} 22 &= 4 \times 5 + 2\ i^{22} &= (i^4)^5 \cdot i^2 = (1)^5 \cdot (-1) = -1 \end{aligned} $$ Write each power of $i$ as $i$, $-1$, $-i$, or $1$. $i^3 \cdot i^{19} =$ $-1$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Powers of i" ] }

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QUESTION IMAGE

Question

Explanation:

Simplify the product of powers

Reduce the exponent modulo 4

Answer: