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Chapter 7: Problem 112

Simplify \({ }^{3} \sqrt{\left(-81 x^{3}\right)}-2 x^{3} \sqrt{3}+5 x^{3} \sqrt{24}\)

Short Answer

Expert verified
The simplified expression is \(-3x + 2x^3\sqrt{3} + 10x^3\sqrt{6}\).

Step by step solution

01

Simplify each term

Let's first simplify each term in the expression: 1. For the first term, we have the cube root of \(-81x^3\): \(\sqrt[3]{(-81x^3)}= -3x\), since the cube root of \(-81\) is \(-3\) and the cube root of \(x^3\) is \(x\). 2. For the second term, we have \(2x^3\) times the square root of 3. We do not need to simplify it further because cube roots and square roots are different. 3. For the third term, we have \(5x^3\) times the square root of \(24\): \(5x^3\sqrt{24} = 5x^3 \cdot (2\sqrt{6}) = 10x^3\sqrt{6}\), since \(24\) can be factored as \((2^3 \cdot 3)\) and the square root of \(2^3\) is \(2\).
02

Combine like terms

Now that we have simplified each term, we can combine any like terms. The simplified expression is: \(-3x + 2x^3\sqrt{3} + 10x^3\sqrt{6}\) There are no like terms in the expression. Therefore, the simplified expression is: \(-3x + 2x^3\sqrt{3} + 10x^3\sqrt{6}\) Since there are no like terms to combine, the expression is already simplified and our final answer is: \(\boxed{-3x + 2x^3\sqrt{3} + 10x^3\sqrt{6}}\).

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Key Concepts

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

Understanding Cube Roots
The cube root of a number is a special value that, when used three times in a multiplication, gives that original number. For example, the cube root of \(-81\) is \(-3\) because \((-3) \times (-3) \times (-3) = -81\).
When dealing with variables like \(x^3\), the cube root of \(x^3\) is simply \(x\). The idea is that taking the cube root "undoes" the cube, leaving you with the base variable.
This is a crucial concept when simplifying algebraic expressions that involve cube roots. For instance, in the expression \( \sqrt[3]{(-81x^3)} \), simplify by considering both the numerical part \(-81\) and the variable part \(x^3\).
Exploring Square Roots
Square roots are another important concept, where you're looking for a number that, when multiplied by itself, gives the original number. For example, the square root of \(4\) is \(2\) because \(2 \times 2 = 4\).
With expressions involving square roots, like \(2x^3\sqrt{3}\), there's often not much more you can do algebraically than simplify numbers under the root if possible. However, square roots remain distinct from other operations, like cube roots, in the same expression.
In our problem, the square root part does not interact directly with cube roots, so each part should be managed separately as shown through our simplification process.
Factoring to Simplify
Factoring involves breaking down a number or an expression into its simplest building blocks or "factors."
In the original problem, to simplify \(5x^3\sqrt{24}\), we factored \(24\) as \(4 \times 6\) or \(2^3 \times 3\).
This allows us to rewrite it as \(5x^3 \times 2\sqrt{6} = 10x^3\sqrt{6}\), by taking the square root of part of the expression. Factoring inside a square root often helps simplify expressions by pulling out perfect squares.
Combining Like Terms
Like terms in algebra are terms whose variables and their exponents match exactly. This allows you to add or subtract these terms since they represent the same type of quantity.
In the given expression, terms like \(-3x\), \(2x^3\sqrt{3}\), and \(10x^3\sqrt{6}\) have different variables or forms and therefore aren't like terms.
To identify like terms, focus on their variable components. Since none of the terms in the simplified expression shares this same structure, the given expression cannot be reduced further by combining.

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