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Chapter 5: Problem 29

Simplify. $$ -1.1 \sqrt{171} $$

Short Answer

Expert verified
-3.3 \sqrt{19}

Step by step solution

01

- Identify the Expression Inside the Square Root

Observe the expression inside the square root. In this case, it is \( 171 \).
02

- Simplify the Square Root

Look for the simplest form of \( \sqrt{171} \). Since 171 is a product of prime factors: \( 171 = 3^2 \times 19 \), rewrite it using properties of square roots: \( \sqrt{171} = \sqrt{3^2 \times 19} = 3 \sqrt{19} \).
03

- Combine Terms Outside the Square Root

Multiply \( -1.1 \) by \( 3 \sqrt{19} \). Therefore, \( -1.1 \sqrt{171} = -1.1 \cdot 3 \sqrt{19} = -3.3 \sqrt{19} \).

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Key Concepts

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

Prime Factorization
Prime factorization is the process of breaking down a composite number into a product of its prime factors. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. To use prime factorization in simplifying square roots, identify the prime factors of the number inside the square root.

For example, in the expression \(\sqrt{171}\), we start by finding the prime factors of 171.

Here's how you do it:
  • 171 is divisible by 3 (because the sum of its digits, 1 + 7 + 1 = 9, is divisible by 3).
  • Divide 171 by 3: 171 梅 3 = 57.
  • 57 is again divisible by 3: 57 梅 3 = 19.
So, the prime factors of 171 are 3, 3, and 19, thus: \ 171 = 3^2 \times 19\.

With the prime factors identified, you can now move to the next step, which is using these factors to simplify the square root.
Properties of Square Roots
Understanding the properties of square roots will help in simplifying expressions efficiently.

Here are some key properties:
  • Square root of a product: \(\sqrt{a \times b} = \sqrt{a} \cdot \sqrt{b} \)
  • Square root of a square: \(\sqrt{a^2} = a \)
Applying these properties to our problem: \(\sqrt{171} = \sqrt{3^2 \times 19}\).
We use the property of the square root of a product:

\(\sqrt{3^2 \times 19} = \sqrt{3^2} \times \sqrt{19} \).
Now, we can simplify \(\sqrt{3^2} \) to 3, making the expression: \(3 \sqrt{19}\).

By understanding and applying these properties, we simplify the square root expression efficiently.
Multiplying Radical Expressions
Multiplying radical expressions involves a few straightforward steps.

Here's what you need to do:
  • Multiply the coefficients (numbers outside the radicals).
  • Multiply the radicands (numbers inside the radicals) if needed.
In our example, the expression is \(-1.1 \sqrt{171}\).
First, simplify the square root term as we did previously, making it \(-1.1 \cdot 3 \sqrt{19}\).
Then, multiply the coefficients:
\(-1.1 \cdot 3 = -3.3\).
As a result, the term outside the radical is now -3.3, keeping the radical part the same. Finally, the simplified form is:
\(-3.3 \sqrt{19}\).

By following these simple steps, you can easily multiply and simplify complex radical expressions.

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