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Chapter 11: Problem 8

For Problems \(1-12\), write each radical in terms of \(i\) and simplify; for example, \(\sqrt{-20}=i \sqrt{20}=i \sqrt{4} \cdot \sqrt{5}=\) \(2 i \sqrt{5}\) $$ \sqrt{-32} $$

Short Answer

Expert verified
The simplified form is \(4i \sqrt{2}\).

Step by step solution

01

Rewrite the Radical in Terms of i

For any negative number under the square root, we can factor out \(-1\) and write \( \sqrt{-n} = \sqrt{-1} \times \sqrt{n}\) which equals \(i \sqrt{n}\) since \(i\) is defined as \( \sqrt{-1}\). So, \( \sqrt{-32} = i \sqrt{32}\).
02

Simplify the Radical

Now, break down \( \sqrt{32}\). First, find the prime factorization of 32, which is \( 32 = 2^5\). Express \( \sqrt{32}\) using these factors: \( \sqrt{32} = \sqrt{4 \times 8} = \sqrt{4} \cdot \sqrt{8}\).
03

Further Simplify Each Factor

Since \( \sqrt{4} = 2 \) and \( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2 \sqrt{2}\), further simplify to get: \( \sqrt{32} = 2 \times 2 \sqrt{2} = 4 \sqrt{2}\).
04

Combine Results to Express the Final Answer

Combine all the simplifications to express the original radical: \( \sqrt{-32} = i \times 4 \sqrt{2}\). Therefore, the simplified form of \( \sqrt{-32}\) is \( 4i \sqrt{2}\).

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Key Concepts

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

Imaginary Unit
The imaginary unit, denoted by the symbol \(i\), is a fundamental concept in complex numbers. It is defined as the square root of \(-1\). This might seem perplexing at first since the square of any real number is never negative. Nonetheless, \(i\) serves as a powerful tool in mathematics:
  • Expression: The expression \(\sqrt{-1} = i\) allows us to work with the square roots of negative numbers.
  • Complex Numbers: Using \(i\), any complex number can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers.
When solving exercises involving square roots of negative numbers, recognizing how to express them as \(i\times\sqrt{n}\) simplifies computations. For instance, in \(\sqrt{-32}\), we transform it into \(i\sqrt{32}\) to clearly separate the negative component. This manipulation paves the way for simplifying radicals accurately.
Simplifying Radicals
Simplifying radicals involves reducing a radical expression to its simplest form. This process requires recognizing and separating perfect squares from other factors. Here's how it is typically done:
  • Identify Factors: Break down the number under the radical into its prime factors. For example, with 32, we have \(32 = 2^5\).
  • Simplify Perfect Squares: Extract perfect squares from the radical. For \(\sqrt{32}\), recognize that \(\sqrt{4} \times \sqrt{8}\) can be simplified as \(2\cdot\sqrt{8}\).
  • Further Simplification: Continue simplifying the factors, where \(\sqrt{8}\) can be rewritten as \(\sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}\).
  • Combine: Lastly, what we have is \(4\sqrt{2}\), a simplified version of \(\sqrt{32}\).
This step-by-step method ensures that radical expressions are simplified efficiently, making further calculations or evaluations easier.
Prime Factorization of Radicals
Prime factorization is a method used to break a number down into its smallest prime factors, which serves as a foundation for simplifying radicals. Let's explore this process:
  • Understanding Prime Factorization: Every integer greater than 1 can be expressed uniquely as a product of prime numbers. For example, \(32 = 2\times2\times2\times2\times2 = 2^5\).
  • Application to Radicals: When simplifying radicals, prime factorization helps identify perfect squares. It makes it easier to separate these perfect squares from the radical part, streamlining simplification.
  • Simplification Through Factorization: In the case of \(\sqrt{32}\), knowing \(32 = 2^5\) helps us rewrite the expression as \(\sqrt{4} \times \sqrt{8}\), simplifying further into \(4\sqrt{2}\) as demonstrated before.
Overall, prime factorization is an excellent strategy for dealing with radicals, particularly when simplification is required. It lays the groundwork for expressing radicals in terms of complex numbers, using the concept of the imaginary unit \(i\).

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