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Chapter 8: Problem 10

Solve. \((5 \sqrt{2})(2 \sqrt{18})\)

Short Answer

Expert verified
60

Step by step solution

01

Simplify Square Roots

Simplify each square root. For \( \sqrt{18} \), we have: \[ \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \sqrt{2} \] Therefore, \(2 \sqrt{18} = 2 \cdot 3 \sqrt{2} = 6 \sqrt{2} \)
02

Multiply the Simplified Expressions

Now we need to multiply \(5 \sqrt{2}\) by \(6 \sqrt{2}\). Using the product of radicals and coefficients: \[ (5 \sqrt{2})(6 \sqrt{2}) = 5 \cdot 6 \cdot \sqrt{2} \cdot \sqrt{2} = 30 \cdot \sqrt{2^2} = 30 \cdot 2 = 60 \]

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Key Concepts

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

Simplifying Square Roots
When working with square roots, it's important to simplify them as much as possible. Simplifying square roots makes calculations easier and solutions clearer. Let's break it down with an example from the problem: \( \sqrt{18} \). To simplify this, you need to find the largest square factor of 18.

Here's the process:
  • Identify the largest perfect square factor of 18, which is 9 (since 9 = 3x3)
  • Express 18 as a product of 9 and another factor: \(18 = 9 \cdot 2 \)
  • Use the property of square roots to separate them: \( \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} \)
  • Since \( \sqrt{9} \) simplifies to 3, the simplified form is \(3 \sqrt{2} \)
This process can be applied to any square root that isn't already in its simplest form.
Multiplying Radicals
Multiplying radicals (or square roots) involves a few fundamental steps. Using the exercise example \((5 \sqrt{2})(2 \sqrt{18})\), we first simplify the square roots.

As we just saw, \(2 \sqrt{18}\) simplifies to \(6 \sqrt{2}\). So the problem becomes \((5 \sqrt{2})(6 \sqrt{2})\).

Follow these steps for multiplying:
  • Multiply the coefficients (the numbers outside the square roots): \(5 \cdot 6 = 30\)
  • Multiply the radicals (the square root values): \( \sqrt{2} \cdot \sqrt{2} = \sqrt{2^2} = \sqrt{4} = 2\)
  • Combine the products from steps 1 and 2: \(30 \cdot 2 = 60\)
So the result of \((5 \sqrt{2})(6 \sqrt{2})\) is 60. Radicals can be tricky, but breaking them down into steps makes it easier.
Algebraic Expressions
Algebraic expressions include numbers, variables, and operations. In your exercise, we used algebraic expressions to solve for a numerical value involving radicals.

Here's a quick refresher on important points when dealing with algebraic expressions:
  • A term like \(5 \sqrt{2}\) is an algebraic expression containing a numerical coefficient (5) and a radical (\sqrt{2})
  • Simplifying square roots before multiplying helps keep both coefficients and radicals manageable
  • Combining like terms, even when radicals are involved, follows the same rules as regular algebra
The exercise is a practical example of how to manipulate and simplify algebraic expressions involving radicals. Practice by solving different problems with similar structures to get comfortable with these concepts.

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