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Chapter 8: Problem 65

Simplify each radical expression. $$ \frac{30 \sqrt{10}}{5 \sqrt{2}} $$

Short Answer

Expert verified
Simplified expression is \( 6\sqrt{5} \).

Step by step solution

01

- Simplify the Fraction

First, separate the constants from the radicals. This gives: \[ \frac{30}{5} \times \frac{\sqrt{10}}{\sqrt{2}} \]
02

- Simplify the Constants

Simplify the fraction with the constants: \[ \frac{30}{5} = 6 \]
03

- Simplify the Radicals

Division under the radicals can be combined: \[ \frac{\sqrt{10}}{\sqrt{2}} = \sqrt{\frac{10}{2}} = \sqrt{5} \]
04

- Multiply the Simplified Parts

Combine the simplified constant with the simplified radical: \[ 6 \times \sqrt{5} = 6\sqrt{5} \]

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Key Concepts

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

Simplifying Fractions
First, let's simplify fractions. A fraction consists of a numerator (the top part) and a denominator (the bottom part). When simplifying fractions, our goal is to reduce the fraction to its simplest form.

To do this, we divide both the numerator and the denominator by their greatest common divisor (GCD). In our example, the original expression is \[ \frac{30 \times \text{some radical}}{5 \times \text{another radical}} \]
We see both 30 and 5 have a common factor of 5. Dividing 30 by 5 gives us 6. Therefore, \[ \frac{30}{5} = 6 \]
Now, the fraction of constants is simplified, making our job much easier!
Simplifying Radicals
Radicals are expressions that contain a square root, cube root, or any higher-order root. Simplifying radicals involves breaking down the expression inside the radical to its simplest form.
In our example, we have: \[ \frac{\text{some radical}}{\text{another radical}} \]
This can be simplified by performing the division within the radicals.
The original radicals are: \[ \frac{\text{some}\root}{\text{another}\root} = \frac{\root{10}}{\root{2}} \]
We combine the radicals under one root as follows: \[ \frac{\root{10}}{\root{2}} = \root{\frac{10}{2}} = \root{5} \]
Now, we have a simplified radical, \root{5}, which makes the expression easier to handle.
Multiplying Radicals
Multiplying radicals may seem daunting, but it's quite straightforward once you grasp the basics.
When multiplying radicals, multiply the numbers outside the radical signs together, and then multiply the numbers inside the radicals together.
In our simplified problem, we already dealt with this separation: \[ 6 \times \root{5} \]
The constant outside the radical is 6, and the simplified radical is \root{5}.
To combine these, simply write them together: \[ 6 \times \root{5} = 6\root{5} \]
There you have it! We've successfully simplified and combined the radical expression.

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