Problem 30 - College Algebra Essentials

Question

Use the quotient rule to simplify the expressions in Exercises \(23-32\) Assume that \(x>0\) $$\frac{\sqrt{24 x^{4}}}{\sqrt{3 x}}$$

Step-by-Step Solution

Answer

The simplified expression is \(2\sqrt{2}x^{3/2}\).

Step 1: Applying the Quotient Rule

Apply the quotient rule \(\sqrt{\frac{24 x^{4}}{3 x}} = \frac{\sqrt{24 x^{4}}}{\sqrt{3 x}}\).

Step 2: Simplify Inside the Square Root

Simplify the fraction inside the square root \(\frac{24 x^{4}}{3 x} = 8x^{3}\).

Step 3: Simplify the Square Root

Simplify the square root \(\sqrt{8x^{3}}\). We split this into \(\sqrt{8} * \sqrt{x^{3}} = 2\sqrt{2}x^{3/2}\).

Key Concepts

Simplifying Radicals

Simplifying radicals means breaking down complicated square roots into simpler parts. This helps in making the expression easier to work with. Consider an expression like \( \sqrt{8} \). The goal in simplifying this is to find its factors that are perfect squares.

• The number 8 can be factored into \( 4 \times 2 \).
• Here, 4 is a perfect square, whose square root is 2, so we have \( \sqrt{4} = 2 \).
• This simplifies \( \sqrt{8} \) to \( 2\sqrt{2} \).

When dealing with expressions that involve variables, the same principle applies. For example, consider \( \sqrt{x^3} \):

• Rewrite \( x^3 \) as \( x^2 \times x \).
• Since \( x^2 \) is a perfect square, \( \sqrt{x^2} = x \).
• This leaves us with \( x\sqrt{x} \) as the simplified form of \( \sqrt{x^3} \).

Understanding these steps is crucial in simplifying any radical expression.

Square Roots

Square roots specifically involve finding the number that, when multiplied by itself, gives the original number. This concept is fundamental in solving equations and simplifying expressions. For numbers like 16, finding the square root is straightforward because \( 16 = 4 \times 4 \), so \( \sqrt{16} = 4 \). However, with numbers like 8, the process involves identifying factors:

• We factor 8 as \( 4 \times 2 \).
• The square root of 4 is 2, giving us \( \sqrt{8} = 2\sqrt{2} \).

When dealing with variable expressions, similar rules apply:Consider \( \sqrt{x^4} \) which represents the square root of a fourth power.

• The expression \( x^4 = (x^2)^2 \).
• Thus, \( \sqrt{x^4} = x^2 \).

For non-perfect squares, you get results like \( \sqrt{x^3} = x^{3/2} \), reflecting both a square and additional root parts.Breaking down these steps helps make calculations involving square roots more manageable and precise.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition, subtraction, multiplication, and division).These expressions can be simplified using various algebraic rules. In our problem, we used division and the square root operation to simplify:

• First, divide the coefficients and subtract the powers of the variables: \( \frac{24x^4}{3x} \).
• This simplifies to \( 8x^3 \), as you divide 24 by 3, and reduce the power of \( x \) by the power it is divided by.
• Next, simplify any radicals present, as seen when √ was taken: simplify \( \sqrt{8x^3} \) to \( 2\sqrt{2}x^{3/2} \).

By understanding the structure and properties of algebraic expressions, you can confidently manipulate and solve them. This includes using the quotient rule, factoring, and applying square roots appropriately.Such knowledge is essential for tackling not just simple expressions, but also more complex algebraic functions and equations.