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Chapter 15: Problem 31

Simplify. $$5 \sqrt{4 x}-3 \sqrt{9 x}$$

Short Answer

Expert verified
\(\sqrt{x}\)

Step by step solution

01

Rewrite the Square Roots

In this expression, rewrite the square roots of 4 and 9 as 2 and 3, respectively. This will give: \(5 \sqrt{4x} - 3 \sqrt{9x}\) becomes \(5*2\sqrt{x} - 3*3\sqrt{x}\)
02

Perform the Multiplication

Now, perform the multiplication of out-front constants with the integers obtained from the square roots: \n\n10* \(\sqrt{x}\) - 9* \(\sqrt{x}\)
03

Combine Like Terms

The next step is to combine like terms, that is, terms containing \(\sqrt{x}\) : \n\n\(10\sqrt{x} - 9\sqrt{x}\) simplifies to \(\sqrt{x}\)

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Key Concepts

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

Like Terms
In algebra, like terms are terms that contain the exact same variables raised to the same power. This similarity allows them to be added or subtracted from each other.
For instance, in the expression available in the exercise, terms like \(10\sqrt{x}\) and \(9\sqrt{x}\) are considered like terms because they both contain the variable \(x\) inside a square root. To combine them, you treat them much like regular numbers:
  • Simply add or subtract the coefficients (the numbers in front).
  • Since \(10\sqrt{x} - 9\sqrt{x}\) simplifies to \(1\sqrt{x}\) or just \(\sqrt{x}\).
This shows the power of recognizing like terms, saving you time and making your answers simpler!
Square Roots
A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2 because \(2 \times 2 = 4\). Similarly, the square root of 9 is 3 because \(3 \times 3 = 9\).
This concept is essential in the exercise because we were required to simplify square roots of products like \(\sqrt{4x}\) and \(\sqrt{9x}\) by first dealing with the numerical part.
The process becomes simpler when you follow these steps:
  • Take the square root of any perfect square outside of the variable. For \(\sqrt{4x}\), the square root of 4 is 2.
  • Multiply this with the coefficient in front of the square root, giving \(5 \times 2\sqrt{x}\).
  • Do the same for the other term: \(3 \times 3\sqrt{x}\).
Thus, knowing how to handle square roots helps in breaking down complex expressions into more manageable pieces.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators (like addition, subtraction, and multiplication). They do not have an equality sign like an equation.
Simplifying algebraic expressions means to reduce them into their simplest form by implementing techniques such as combining like terms and solving for components like square roots.
  • Identify terms that can be combined (like terms) to streamline the expression.
  • Apply arithmetic operations systematically across numbers and variables to reach a simplified form.
From the original exercise, simplification was key in reducing \(5\sqrt{4x} - 3\sqrt{9x}\) to just \(\sqrt{x}\), offering a clearer, minimal representation of the algebraic expression.

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