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Chapter 8: Problem 10

Write in terms of \(i\) and then simplify. \(\sqrt{-25} \cdot \sqrt{-1}\)

Short Answer

Expert verified
The simplified expression is -5.

Step by step solution

01

Express with Imaginary Unit

The imaginary unit is represented by \(i\), where \(i = \sqrt{-1}\). For \(\sqrt{-25}\), recognizing it as \(\sqrt{25 \times -1}\) helps simplify it to \(\sqrt{25} \cdot \sqrt{-1}\). Hence, \(\sqrt{-25} = 5i\).
02

Write the Expression in Terms of \(i\)

Given the expression \(\sqrt{-25} \cdot \sqrt{-1}\), substitute \(\sqrt{-25}\) with \(5i\) and \(\sqrt{-1}\) with \(i\). This gives \(5i \cdot i\).
03

Simplify the Expression

Simplify \(5i \cdot i\) by using the property of powers of \(i\), which is \(i^2 = -1\). Thus, \(5i \cdot i = 5i^2 = 5 \times -1 = -5\).

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Key Concepts

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

Imaginary Unit
In the fascinating world of numbers, the imaginary unit, denoted as \(i\), plays a crucial role when dealing with square roots of negative numbers. The imaginary unit is essential because it allows us to expand the number system beyond what is possible with only real numbers. It is defined by the relation \(i = \sqrt{-1}\).

When you take the square root of a negative number, such as \(-25\), the imaginary unit comes in handy. You can express \(\sqrt{-25}\) as \(\sqrt{25 \times -1}\), which separates into \(\sqrt{25} \cdot \sqrt{-1}\).
  • \(\sqrt{25}\) equals 5 because 5 is the number that, when multiplied by itself, gives 25.
  • \(\sqrt{-1}\) is, by definition, \(i\).
Thus, \(\sqrt{-25} = 5i\). This process highlights how powerful the imaginary unit is, effectively handling the square roots of negative numbers.
Simplifying Complex Expressions
Simplifying complex expressions involving imaginary numbers can initially seem daunting, but it becomes straightforward once you grasp the fundamental concepts. Using the imaginary unit \(i\), any complex expression can be altered into a simplified form.

Taking an expression like \(\sqrt{-25} \cdot \sqrt{-1}\) as an example, you start by substituting known values where \(\sqrt{-25}\) becomes \(5i\) and \(\sqrt{-1}\) becomes \(i\). Thus, the expression simplifies to \(5i \cdot i\).
  • This step essentially converts a seemingly complex problem into a simple multiplication task with imaginary numbers.
  • Combining like terms and making use of the properties of \(i\) leads to further simplification.
Simplifying involves multiplying \(5i\) by \(i\) to get \(5i^2\). This can be further simplified using the powers of \(i\).
Powers of i
The powers of \(i\) follow a fascinating yet straightforward cyclic pattern. Knowing this pattern is key when working with imaginary numbers.

The cycle begins with the simple definition of \(i\):
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)
Once you reach \(i^4\), the cycle repeats, which means \(i^5 = i\), \(i^6 = -1\), and so on.

By understanding this cycle, you can easily simplify expressions involving higher powers of \(i\). For instance, in the expression \(5i^2\), substitute \(i^2\) with \(-1\) to get \(5 \times -1 = -5\).
These shortcuts allow you to quickly and accurately simplify complex expressions, thanks to the periodic nature of \(i\)'s powers.

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Most popular questions from this chapter

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