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Chapter 8: Problem 7

Simplify. $$9 \sqrt{40}$$

Short Answer

Expert verified
The simplified form of \(9 \sqrt{40}\) is \(18 \sqrt{10}\)

Step by step solution

01

Simplify the square root

Find factors of 40 that are perfect squares. 40 can be written as 4 times 10, and 4 is a perfect square. Therefore, \(\sqrt{40} = \sqrt{4} \times \sqrt{10} = 2 \times \sqrt{10}\)
02

Multiply with the integer

Multiply the found square root with the integer outside the root. So, \(9 \times 2 \sqrt{10} = 18 \sqrt{10}\)

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Key Concepts

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

Perfect Squares
Perfect squares are numbers that can be expressed as the product of an integer with itself. Understanding perfect squares is crucial for simplifying square roots. For instance, numbers like 1, 4, 9, 16, 25, etc., are perfect squares because you can write them as \(1\times1\), \(2\times2\), \(3\times3\), and so on. Knowing these helps you break down more complex square roots.
  • When simplifying a square root, the goal is to find a factor that is a perfect square because it can be taken out from under the square root sign.
  • For example, to simplify \(\sqrt{40}\), identify the largest perfect square that divides 40, which is 4, because \(4 \times 10 = 40\).
  • The square root of 4 is 2, so \(\sqrt{40}\) can be simplified to \(2\sqrt{10}\).
Recognizing and factoring out perfect squares makes square root expressions simpler.
Multiplying Square Roots
Multiplying square roots involves understanding how to deal with expressions both inside and outside of the square root. When you have a term like \(9\sqrt{40}\), and you've simplified the square root to \(\sqrt{10}\), you can multiply any numbers outside of the square root first.
  • From the previous section, we know \(\sqrt{40} = 2\sqrt{10}\).
  • The integer outside the square root here is 9. Therefore, multiply 9 by the 2, giving us \(18\). This will be the coefficient for your square root.
  • The expression becomes \(18\sqrt{10}\).
All operations involving square roots and whole numbers should follow the same order of operation principles as regular arithmetic.
Factorization
Factorization is the process of breaking down numbers into their prime factors. This method is especially useful when simplifying square roots because it helps identify perfect squares.
  • To factorize 40, the number under our root, you break it into its prime factors: \(2 \times 2 \times 2 \times 5\).
  • The factorization aids in identifying the perfect square, which in our case is \(2 \times 2 = 4\).
  • Here, 4 is a perfect square, making it easier to simplify the square root to \(2\sqrt{10}\).
Factorizing numbers is a critical step in simplifying expressions and equations involving roots, providing a systematic way to resolve them effectively.

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Most popular questions from this chapter

The shorter diagonal of a rhombus with a \(70^{\circ}\) angle is \(122 \mathrm{cm}\) long. How long, to the nearest centimeter, is the longer diagonal?

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