Problem 80 Write the number in $i$-form. ... [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 80

Write the number in $i$-form. $$\sqrt{-\frac{20}{49}}$$

Short Answer

Expert verified

So, the number in the $i$ form is $\frac{2\sqrt{5}}{7}i$.

Step by step solution

Identify the Negative Square Root

The given number is $\sqrt{-\frac{20}{49}}$. The square root is taken of a negative number.

Convert the Negative Square Root

Since the square root is negative, the imaginary unit $i$ is used. You can write: $\sqrt{-\frac{20}{49}} = \sqrt{\frac{20}{49}}i = \sqrt{\frac{4}{7} * \frac{5}{7}}i$.

Calculate the final i-form

Further simplify to get: $\sqrt{\frac{4}{7} * \frac{5}{7}}i = \sqrt{\frac{4}{7}} * \sqrt{\frac{5}{7}}i = \frac{2}{\sqrt{7}} * \sqrt{\frac{5}{7}}i = \frac{2\sqrt{5}}{7}i$.

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Key Concepts

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

Square Roots

When we talk about square roots, we're discussing a special type of mathematical operation. Imagine you have a number, like 9. The square root of 9 is a number that, when multiplied by itself, gives you back 9. In this case, the answer is 3 because $3 \times 3 = 9$. But what happens when we deal with fractions or other complex numbers? Let's consider the square root of a fraction, such as $\sqrt{\frac{25}{100}}$. To solve this, you can take the square root of the numerator (25) and the denominator (100) separately, which gives you $\frac{5}{10}$, or $\frac{1}{2}$ after simplification. Always remember:

The square root symbol $\sqrt{}$ indicates the principal, or positive, square root.
If you are dealing with perfect squares, like 4 or 16, they split evenly into whole numbers.
For fractions, simplify the square root of both top and bottom numbers separately.

Negative Numbers

Negative numbers are anything less than zero. They are often represented with a minus sign (-). Think of them as points below zero on a number line. In many real-life situations, they can represent a loss or some form of removal. What鈥檚 curious is when negative numbers are used under a square root symbol. Normally, the square root of a number is not negative because multiplying two negatives gives you a positive number. But if you see a square root symbol with a negative number inside, you might wonder how to handle it. Here's some insight:

A negative number inside a square root leads us to complex or imaginary numbers.
We need to use a trick: rewrite that negative value as a product involving the imaginary unit $i$.
$i$ stands for $\sqrt{-1}$, helping us to deal with these operations.

This means $\sqrt{-9}$ becomes $3i$, switching from a real to an imaginary number.

Imaginary Unit

We use imaginary numbers to make sense of square roots of negative numbers. The key player here is the imaginary unit, $i$. It's defined as $\sqrt{-1}$. This concept helps us solve many problems that involve taking square roots of negative numbers. Let's see how $i$ works in action:

If you take $i^2$, you get $-1$ because it's defined as $i \times i\ = \sqrt{-1} \times \sqrt{-1}\ = -1$.
Using this, you can easily handle expressions like $\sqrt{-25}$, which become $5i$.
The imaginary unit is not about fantasy, but a necessity in mathematics, physics, and engineering.

Whenever you see $\sqrt{-x}$, you should think $i\sqrt{x}$. This transforms our previous challenge into a simpler form, ready for further calculation.

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91影视

Write the number in \(i\)-form. $$\sqrt{-\frac{20}{49}}$$

Short Answer

Step by step solution

Identify the Negative Square Root

Convert the Negative Square Root

Calculate the final i-form

Key Concepts

Square Roots

Negative Numbers

Imaginary Unit

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