Problem 34 Find the indicated powers of com... [FREE SOLUTION]

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Chapter 7: Problem 34

Find the indicated powers of complex numbers. $$(5 i)^{2}$$

Short Answer

Expert verified

-25

Step by step solution

Identify the base and the exponent

The base is the complex number in the expression $5i$. The exponent is $2$.

Use the property of exponents

To find the power of a complex number, we can use the property \[ (a \, b)^n = a^n \, b^n \]. Apply this property to $ (5i)^{2} $.

Compute the power

Calculate $ 5^{2} $ and $ i^{2} $ separately: \ 5^2 = 25 \ and \ i^2 = -1 \. Then multiply the results: \ 25 \, \ -1 = -25 \.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers

Complex numbers can seem confusing, but they are just an extension of regular numbers that include the imaginary unit, denoted as $ i $.
An imaginary unit $ i $ is defined by the property $ i^2 = -1 $.
Combining real and imaginary parts gives us a complex number, typically written in the form:$ a + bi $, where $a$ and $b$ are real numbers.
For example, in the complex number $ 5i $, $ a = 0 $ and $ b = 5 $.
Here, the real part $ 0 $ is often omitted.
This extension allows us to solve equations that do not have solutions in the realm of real numbers alone.

Exponents

Exponents are a way of representing repeated multiplication of a number by itself.
The expression $ x^n $ indicates that the number $ x $ is multiplied by itself $ n $ times.
There are several rules for working with exponents that help simplify calculations.
For example, when multiplying two exponential numbers with the same base, we add their exponents: $ x^a \times x^b = x^{a+b} $.
Another rule is that any number raised to the power of zero is 1: $ x^0 = 1 $.
These rules apply to both real and complex numbers.

Properties of Exponents

Understanding the properties of exponents is crucial when dealing with both real and complex numbers.
Some key properties include:

Product of Powers: $ a^m \times a^n = a^{m+n} $

Power of a Power: $(a^m)^n = a^{mn} $

Quotient of Powers: $ \frac{a^m}{a^n} = a^{m-n} $ \ where $ a eq 0 $

Applying these properties to complex numbers follows the same rules.
When given an expression like $ (5i)^2 $, we break it down using these properties:
First, we separate the real and imaginary parts, and then apply the rules to each part accordingly.
So $ (5i)^2 $ becomes $ 5^2 \times i^2 $, which simplifies to $ 25 \times -1 = -25 $.
This consistency makes working with complex numbers more manageable.

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91影视

Find the indicated powers of complex numbers. $$(5 i)^{2}$$

Short Answer

Step by step solution

Identify the base and the exponent

Use the property of exponents

Compute the power

Key Concepts

Complex Numbers

Exponents

Properties of Exponents

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