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Chapter 9: Problem 34

Simplify the expression. $$4 a^{2} \sqrt{18 a}+7 \sqrt{200 a^{5}}$$

Short Answer

Expert verified
The simplified expression is \(82a^{2}\sqrt{2a}\).

Step by step solution

01

Simplify the square roots

First, simplify each square root separately. \(\sqrt{18a} = \sqrt{9 \cdot 2 \cdot a} = 3\sqrt{2a}\) and \(\sqrt{200a^{5}} = \sqrt{100 \cdot 2 \cdot a^{4} \cdot a} = 10a^{2}\sqrt{2a}\)
02

Substitute simplified square roots into the expression

Now you can replace the simplified square roots back into the original equation: \(4a^{2} \cdot 3\sqrt{2a} + 7 \cdot 10a^{2}\sqrt{2a} = 12a^{2}\sqrt{2a} + 70a^{2}\sqrt{2a}\)
03

Combine like terms

The terms in the expression are now like and can be combined: \(12a^{2}\sqrt{2a} + 70a^{2}\sqrt{2a} = 82a^{2}\sqrt{2a}\)

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Key Concepts

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

Simplifying Expressions
Algebra is full of expressions that need simplifying. When you simplify an expression like the one in this exercise, you are making it as neat as possible while keeping its value. This process helps you better understand what the expression really means.
You often start by breaking down complex numbers and variables into smaller, easier parts.
  • Look for opportunities to factor numbers or expressions.
  • Deal with square roots by pulling out perfect squares first.
  • Aim for a simple expression that's easy to work with in further calculations.
By simplifying, you make the expression less intimidating and more approachable. This will help you see patterns and potential solutions more quickly as you advance in algebra.
Square Roots
Square roots can make expressions more complicated but also more interesting! When you see a square root sign (鈭�), you're dealing with the number that, when multiplied by itself, gives you the original number. In our exercise, we dealt with square roots such as \(\sqrt{18a}\) and \(\sqrt{200a^5}\). Breaking these down is your first step.
  • Check if the number under the root can be factored into a perfect square.
  • A perfect square is a number like 9 or 100, as they are 3虏 and 10虏, respectively.
  • Taking out the perfect square simplifies your root tremendously.
As we saw:
\(\sqrt{18a} = 3\sqrt{2a}\) since \(18a = 9 \times 2a\).
\(\sqrt{200a^5} = 10a^2\sqrt{2a}\) as \(200a^5 = 100 \times 2a^4 \times a\).
By understanding square roots, simplifying them becomes much easier!
Combining Like Terms
"Like terms" are parts of an algebraic expression that have the same variables raised to the same power. In the final step of simplification, combining these terms is crucial. This is because it reduces the expression to its simplest form, making it easier to work with in subsequent mathematics.
  • Identify terms that have the same variable parts, such as \(a^2\sqrt{2a}\).
  • Add or subtract the coefficients (the numbers in front) of these like terms.
  • Ensure the new coefficient is paired with the shared variable part.
In the exercise given, after simplifying square roots, the terms \(12a^2\sqrt{2a}\) and \(70a^2\sqrt{2a}\) were combined:
\(12a^2\sqrt{2a} + 70a^2\sqrt{2a} = 82a^2\sqrt{2a}\).
Mastering this step ensures that you see algebra not as a mess of numbers and letters, but as a streamlined expression you can easily understand and manipulate!

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