Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 99

Chapter 2, Problem 99

Simplify each power of i. 1/i^-11

Verified step by step guidance

Recall that the imaginary unit $i$ has the property $i^2 = -1$, and powers of $i$ cycle every 4 steps: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, and then the pattern repeats.

Rewrite the expression $\frac{1}{i^{-11}}$ as $i^{11}$ because dividing by $i^{-11}$ is the same as multiplying by $i^{11}$ (using the property $\frac{1}{a^{-n}} = a^n$).

To simplify $i^{11}$, find the remainder when 11 is divided by 4, since powers of $i$ repeat every 4: calculate $11 \mod 4$.

Use the remainder from the previous step to rewrite $i^{11}$ as one of $i^0$, $i^1$, $i^2$, or $i^3$, which correspond to $1$, $i$, $-1$, or $-i$ respectively.

Express the simplified form of $i^{11}$ based on the cycle and write the final simplified expression.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Imaginary Unit and Powers of i

The imaginary unit i is defined as the square root of -1, with the property i² = -1. Powers of i cycle every four exponents: i¹ = i, i² = -1, i³ = -i, and i⁴ = 1, then the pattern repeats. Understanding this cycle helps simplify expressions involving powers of i.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, a⁻ⁿ = 1/aⁿ. Applying this rule allows rewriting expressions like i⁻¹ as 1/i, which is essential for simplifying powers with negative exponents.

Simplifying Complex Fractions

Simplifying expressions involving fractions with powers, such as 1/i⁻¹¹, requires applying exponent rules and properties of i. This often involves rewriting negative exponents, using the cyclic nature of i's powers, and reducing the expression to a standard form without imaginary units in the denominator.

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Use the method described in Exercises 83–86, if applicable, and properties of absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can be solved by inspection.)

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Solve each equation. x^-2/3+x^-1/3-6=0

Simplify each power of i. 1/i-11

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Key Concepts

Imaginary Unit and Powers of i

Negative Exponents

Simplifying Complex Fractions

Simplify each power of i. 1/i^-11