Problem 119 - Elementary and Intermediate Algebra

Question

Explain why each expression is not in simplified form. a. \(\sqrt[3]{9 x^{4}}\) b. \(\sqrt{\frac{24 m}{25}}\) c. \(\frac{\sqrt[4]{c^{3}}}{\sqrt[4]{16}}\)

Step-by-Step Solution

Answer

Each expression contains factors that can be further simplified by separating perfect powers or roots.

Step 1: Identifying Simplifiable Parts in Expression a

Expression a is \(\sqrt[3]{9 x^{4}}\). Here, the term \(9\) can be factored as \(3^2\), and \(x^{4}\) can be rewritten as \(x^3 \cdot x\). In a simplified cube root expression, any factor's power should be less than 3, but here \(x^3\) can be taken out of the cube root as \(x\). This suggests that the expression is not simplified.

Step 2: Identifying Simplifiable Parts in Expression b

The expression is \(\sqrt{\frac{24 m}{25}}\). For a radical to be simplified, any perfect square factors should be extracted. The numerator \(24m\) can be factored into \(4 \cdot 6 \cdot m\), and \(\sqrt{4} = 2\), which needs to be taken out of the square root. The denominator \(25\) is already a perfect square \((5^2)\) and can remain outside the radical. Since \(\sqrt{4}\) was not simplified directly as \(2\), the expression is not in simplified form.

Step 3: Identifying Simplifiable Parts in Expression c

For the expression \(\frac{\sqrt[4]{c^{3}}}{\sqrt[4]{16}}\), both the numerator and denominator can potentially be simplified further. The term \(c^3\) is not a perfect fourth power. However, the denominator \(16\) equals \(2^4\), and the fourth root of a perfect fourth power should be computed. Thus, the expression is not simplified because \(\sqrt[4]{16}\) should be simplified to \(2\). Also, any simplification of the numerator needs to be attempted.

Step 4: Summary Simplifications for All Expressions

Based on the analyses: - For expression a: \(x^{3} = (9x)\)'s cube should be extracted as \(x\cdot \sqrt[3]{9x}\). - For expression b: The factor \(4\) should be simplified as \(2\) outside the square root. - For expression c: \(\sqrt[4]{16}\) should simplify to \(2\), altering the expression to \(\frac{\text{some function of } c}{2}\). Each expression contains elements that can be further reduced.

Key Concepts

Cube Roots

Understanding cube roots is crucial for simplifying algebraic expressions like \(\sqrt[3]{9x^4}\). A cube root, denoted by \(\sqrt[3]{x}\), represents a number which, when multiplied by itself three times, equals \(x\). For instance, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\).
In algebra, to simplify an expression like \(\sqrt[3]{9x^4}\), we need to extract perfect cubes.
- First, recognize \(9\) as \(3^2\), which is not a perfect cube, so it will remain inside the cube root.- Second, identify \(x^4\) as \(x^3 \cdot x\). Since \(x^3\) is a perfect cube, it can be taken out of the cube root as \(x\).
The simplified form becomes \(x \cdot \sqrt[3]{9x}\), showing the importance of isolating and extracting cube factors.

Square Roots

Square roots are operations used to find a number which, when multiplied by itself, gives the original number. The square root is represented by \(\sqrt{x}\). A common example would be \(\sqrt{9} = 3\) because \(3^2 = 9\).
For a given expression such as \(\sqrt{\frac{24m}{25}}\), simplifying involves identifying and extracting perfect square factors.
- Decompose the numerator as \(4 \cdot 6 \cdot m\). The \(4\) is a perfect square and can be simplified to \(2\), allowing it to be extracted outside the square root.- The denominator, \(25\), is a perfect square \((5^2)\) and should remain outside the root as \(5\).
So, the simplified version becomes \(\frac{2\sqrt{6m}}{5}\). Handling square roots this way helps achieve a tidier expression and better understanding.

Fraction Simplification

Simplifying fractions involves reducing them to their smallest form, making calculations easier. Consider the expression \(\frac{\sqrt[4]{c^3}}{\sqrt[4]{16}}\). Simplification requires evaluating both the numerator and denominator.
- The denominator \(\sqrt[4]{16}\) simplifies directly to \(2\) because \(16 = 2^4\).- The numerator, \(\sqrt[4]{c^3}\), requires attempting simplification. While \(c^3\) is not a complete fourth power, exploring any simplifiable factors is crucial.
The fraction simplifies to \(\frac{\sqrt[4]{c^3}}{2}\) by focusing on readily computable radical simplifications.

Radicals

Radicals encompass square roots, cube roots, and more, serving as handy tools in expressing root functions. A radical is typically shown as \(\sqrt[n]{x}\) for "n-th" roots. Simplifying radicals is about identifying perfect powers within the radicand and factoring them out.
To consider why \(\sqrt[3]{9x^4}\), \(\sqrt{\frac{24m}{25}}\), and \(\frac{\sqrt[4]{c^3}}{\sqrt[4]{16}}\) aren't simplified, it's vital to:

• Recognize perfect powers (e.g., \(x^3\), \(4\), \(16\)) within the expressions.
• Extract these perfect powers to reduce expression complexity.

Simplifying radicals involves consistent practice in identifying these factors, ensuring clarity and ease in expression handling.