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Chapter 8: Problem 41

Simplify. All variables represent positive values. $$\sqrt{32 x^{5}}-\sqrt{18 x^{5}}$$

Short Answer

Expert verified
The simplified form of the expression is \(x^{2}\sqrt{2x}\)

Step by step solution

01

Simplify the individual square roots

The two square roots can be simplified to their simplest form. The square root of \(32x^{5}\) is \(4x^{2}\sqrt{2x}\) and the square root of \(18x^{5}\) is \(3x^{2}\sqrt{2x}\). The expression becomes \(4x^{2}\sqrt{2x} - 3x^{2}\sqrt{2x}\).
02

Factor out common factors

Next, factor out the common factor from the two terms to simplify the expression further. In this case, the common factor is \(x^{2}\sqrt{2x}\). Factoring out gives \(x^{2}\sqrt{2x}(4 - 3)\).
03

Simplify the expression

Now simplify the expression further by evaluating the bracket. This gives \(x^{2}\sqrt{2x}(1)\) which further simplifies to \(x^{2}\sqrt{2x}\).

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Key Concepts

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

Square Roots
Square roots are all about discovering what number, when multiplied by itself, equals the given value. For instance, the square root of 9 is 3, because 3 multiplied by 3 gives 9. When simplifying square roots, we aim to bring them down to their simplest form for easier calculations.

Let's break it down with the example in the exercise:
  • The square root of 32 can be simplified by identifying that 32 is equal to 16 times 2. Since 16 is a perfect square, \( \sqrt{16} = 4 \,\) we can simplify \( \sqrt{32} = 4\sqrt{2} \).
  • Similarly, for the square root of 18, which equals 9 times 2, since 9 is a perfect square, \( \sqrt{9} = 3 \), we further simplify to get \( \sqrt{18} = 3\sqrt{2} \).
  • For expressions like \( \sqrt{32x^5} \), the properties of exponents are handy. Since \( \sqrt{x^5} = x^{5/2} \,\) it separates into \( x^2\sqrt{x}\sqrt{2} \,\) where each part can be handled more simply.
By learning these steps, simplifying square roots becomes a straightforward process.
Factoring
Factoring is the process of breaking down an expression into smaller or more manageable parts, or factors, that can be multiplied to give the original expression. This process makes expressions simpler, especially when dealing with algebraic problems.

In the original exercise, factoring was used to pull out the common elements from the simplified terms of square roots. Here鈥檚 how it works:
  • From the expression \( 4x^2\sqrt{2x} - 3x^2\sqrt{2x} \,\) notice both terms contain a common factor, \( x^2\sqrt{2x} \).
  • Factor this common part outside the parentheses: \( x^2\sqrt{2x}(4 - 3) \).
  • Then simplify inside the parentheses: \( 4 - 3 = 1 \).
By using factoring here, the expression collapses into a much simpler form: \( x^2\sqrt{2x} \,\) making it easier to work with or interpret.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the basis for algebra, allowing us to represent real-world problems in mathematical terms.

In the exercise, the expression \( \sqrt{32x^5} - \sqrt{18x^5} \) denotes an algebraic union of two terms. Here, what's important is how you manipulate these expressions:
  • Recognize that variables within square roots, like \( x^5 \,\) can be broken down using the rules of exponents. This helps simplify the terms significantly.
  • Understand that simplifying each expression independently makes it easier to handle the full equation.
  • Use operations like subtraction or addition only after expressions have been simplified, leveraging factored components when necessary.
Algebraic expressions, although they might seem complex, follow systematic rules that let us untangle and understand them clearly.

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Most popular questions from this chapter

Use scientific notation to simplify \(\sqrt{0.00000004}\).

Is \((-4)^{1 / 2}\) a real number? Explain.

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