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Chapter 8: Problem 10

Simplify. $$\sqrt{5} \cdot \sqrt{35}$$

Short Answer

Expert verified
The simplification of \( \sqrt{5} \cdot \sqrt{35} \) is \( 5\sqrt{7} \).

Step by step solution

01

Multiply the numbers within the square roots

Multiply the numbers within the square roots, the result is \( \sqrt{5} \cdot \sqrt{35} = \sqrt{5 \cdot 35} \)
02

Simplify the number within the square root

Now, simplify the number under the square root. 5 multiplied by 35 equals 175. So, \( \sqrt{5 \cdot 35} = \sqrt{175} \)
03

Further simplify the square root

The number 175 can be expressed as the product of 7 and 25, where 25 is a perfect square. Hence we can simplify further as: \( \sqrt{175} = \sqrt{7 \cdot 25} \) = \( \sqrt{7} \cdot \sqrt{25} = 5 \cdot \sqrt{7} \)

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Key Concepts

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

Multiplication of Radicals
When you multiply radicals, it's like multiplying any other numbers, but with an extra step. Radicals are simply the square roots of numbers. To multiply them, you just multiply the numbers inside the square roots, as long as they have the same index. This is called the property of radicals, which says:
  • \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \)
For example, if you have \( \sqrt{5} \cdot \sqrt{35} \), you multiply the numbers inside to get \( \sqrt{5 \cdot 35} \), which equals \( \sqrt{175} \). Remember, you must always ensure the indices are the same before multiplying. This step sets you up for simplification.
Simplification of Radicals
Simplifying radicals involves making the numbers under the square root sign as simple as possible. In the example \( \sqrt{175} \), the goal is to break it down into simpler parts. Start by factoring the number under the square root. For 175, it's \( 7 \times 25 \). Find any factors that are perfect squares, as these can be taken out from under the radical. Here, 25 is a perfect square because \( \sqrt{25} = 5 \). So, we can express this as:
  • \( \sqrt{175} = \sqrt{7 \cdot 25} \)
  • \( = \sqrt{7} \cdot \sqrt{25} \)
  • \( = 5 \cdot \sqrt{7} \)
By doing this, you simplify the radical, which makes the number easier to work with. Look for other factors if possible to completely simplify your expression.
Perfect Squares
A perfect square is a number that can be expressed as the square of an integer. Some common perfect squares are 4, 9, 16, 25, 36, and so on. Recognizing them is key to simplifying radicals. When simplifying an expression like \( \sqrt{175} \), identifying the 25 as a perfect square is crucial.
  • \( \sqrt{25} = 5 \)
This means \( 25 \) can move outside of the radical, simplifying your expression. Perfect squares help make complex radical expressions manageable. If you can find a perfect square factor in a number under a radical, then split it out to simplify easily. Spotting these can come with practice, so it鈥檚 important to familiarize yourself with common perfect squares and use them often.

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