Problem 10 Determine whether the complex nu... [FREE SOLUTION]

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Elementary Algebra Within Reach

Ron Larson, Kimberly Nolting

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 10: Problem 10

Determine whether the complex numbers are equal. $\sqrt{16}+\sqrt{-9}$ and $4-3 i$

Short Answer

Expert verified

No, the complex numbers $\sqrt{16}+\sqrt{-9}$ and $4-3i$ are not equal.

Step by step solution

Evaluate the square roots

First, solve individual square roots. The square root of a positive number is real, and the square root of a negative number is imaginary. The square root of 16 is 4. The square root of -9 is 3i because we know that $\sqrt{-1} = i $, so $\sqrt{-9} = \sqrt{9} . \sqrt{-1} = 3i $

Sum the result of the square roots

We add the outcome from the first step. So, $\sqrt{16}+\sqrt{-9} = 4 + 3i$

Comparing the results

Now compare the results of both expressions. $4+3i$ equals $4-3i$ when compared symbol by symbol which is not true as the imaginary parts (+3i and -3i) do not equal each other. Hence, the two expressions are not equal.

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Key Concepts

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

Square Roots

The square root is a mathematical operation that finds a value which, when multiplied by itself, gives the original number. Square roots can be applied to both positive and negative numbers, each yielding a distinct result.

For positive numbers, the square root is straightforward. For example, \[\sqrt{16} = 4\] This is because 4 multiplied by itself gives 16.
Negative numbers involve imaginary numbers when taking their square root, which we discuss in the next section.

Understanding square roots is essential because it forms the basis of simplifying complex numbers. In the problem at hand, calculating these roots separately was the initial step in solving whether two complex numbers are equal.

Imaginary Numbers

Imaginary numbers are a crucial concept when dealing with square roots of negative numbers. Normally, square roots of negative numbers do not exist in the real number system. Here, the imaginary unit, denoted as $i$, becomes important. It's defined by the equation:\[ i^2 = -1 \]This provides a way to express square roots of negative numbers.

The imaginary unit $i$ signifies the square root of -1.
For example, to find the square root of -9, you can express it as:\[\sqrt{-9} = \sqrt{9} \times \sqrt{-1} = 3i\]which means 3 times the imaginary unit $i$.

This way, math can be extended to handle solutions beyond real numbers. Imaginary numbers are crucial for complex number calculations such as those in the original problem.

Equality of Complex Numbers

Complex numbers combine real numbers with imaginary numbers to form expressions like $a + bi$, where $a$ is the real part and $bi$ is the imaginary part. For complex numbers to be equal, both their real and imaginary components must be identical.

Two complex numbers $a + bi$ and $c + di$ are equal only if $a = c$ and $b = d$.
In our problem, comparing $4 + 3i$ and $4 - 3i$, though the real part (4) is the same, the imaginary parts ($+3i$ and $-3i$) are different.

This fundamental rule aids significantly in recognizing when complex numbers should not be considered equal or when errors might have been made in a calculation. Knowing this helps solve and understand problems such as the current exercise more effectively.

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91影视

Determine whether the complex numbers are equal. \(\sqrt{16}+\sqrt{-9}\) and \(4-3 i\)

Short Answer

Step by step solution

Evaluate the square roots

Sum the result of the square roots

Comparing the results

Key Concepts

Square Roots

Imaginary Numbers

Equality of Complex Numbers

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