Problem 19 - Algebra 2 and Trigonometry

Question

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{\frac{3 b^{3}}{44}} $$

Step-by-Step Solution

Answer

Simplified form: \( \frac{b \sqrt{3b}}{2 \sqrt{11}} \).

Step 1: Simplify the Fraction Inside the Radical

Start by simplifying the fraction inside the square root. We have \( \frac{3b^3}{44} \). Since 3 and 44 are coprime, we cannot simplify the coefficients any further. Therefore, the fraction remains as \( \frac{3b^3}{44} \).

Step 2: Apply the Square Root to the Fraction

The square root of a fraction can be separated into a quotient of two square roots: \( \sqrt{\frac{3b^3}{44}} = \frac{\sqrt{3b^3}}{\sqrt{44}} \).

Step 3: Simplify the Numerator

Simplify \( \sqrt{3b^3} \): Factor \( b^3 \) as \( b^2 \times b \). This allows us to take \( b^2 \) out of the square root since \( \sqrt{b^2} = b \): \( \sqrt{3b^3} = b \sqrt{3b} \).

Step 4: Simplify the Denominator

Simplify \( \sqrt{44} \): Factor 44 as \( 4 \times 11 \). Since \( 4 = 2^2 \), we can take 2 out of the square root: \( \sqrt{44} = 2 \sqrt{11} \).

Step 5: Write the Simplified Radicals in Fraction Form

Combine the results from steps 3 and 4 and write them in fraction form: \( \frac{b \sqrt{3b}}{2 \sqrt{11}} \).

Key Concepts

Simplifying Fractions

When simplifying fractions, our main goal is to reduce the fraction to its simplest form. This means finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this number.

• For example, in the fraction \( \frac{6}{8} \), both numbers can be divided by 2, which is their GCD, resulting in \( \frac{3}{4} \).
• It is crucial to remember that reducing fractions does not alter their value; it merely makes them easier to work with.
• In cases of fractions with variables, ensure that any variable expressions are not zero before simplifying.

In the given exercise, \( \frac{3b^3}{44} \), 3 and 44 are coprime, meaning they have no common divisors other than 1. Thus, the fraction remains the same without further simplification needed.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. Taking a square root is essentially the opposite of squaring a number.

• A perfect square, like 9, has an integer square root, such as 3, because \(3 \times 3 = 9\).
• Some numbers don't have integer square roots, so they remain in their radical form, like \( \sqrt{2} \).
• When a square root is applied to an entire fraction, distribute the square root to both the numerator and denominator separately. For example, \( \sqrt{\frac{9}{16}} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4} \).

In the exercise, the expression \( \sqrt{\frac{3b^3}{44}} = \frac{\sqrt{3b^3}}{\sqrt{44}} \), showcases how square roots of fractions are handled.

Radicals

Radicals, often represented by the square root symbol \( \sqrt{} \), can simplify complex expressions. A primary focus is on understanding how to manipulate these to simplify expressions.

• Recognize that not all numbers need simplification; finding perfect squares within the radicand (the number under the square root) is key.
• Breaking down the radicand into factors can help identify which parts of a number can be simplified, such as taking \( \sqrt{18} \) can be simplified to \( 3\sqrt{2} \) by realizing \( 18 = 9 \times 2 \), and 9 is a perfect square.
• For variables, like \( b^3 \) becomes \( b^2 \times b \); here, \( \sqrt{b^2} = b \), which can be taken out of the radical.

The usage of radicals in our example transformed \( \sqrt{3b^3} \) into \( b \sqrt{3b} \), simplifying the expression by tackling root elements separately.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations. When simplifying these, ensure to handle each component carefully:

• Expressions like \( 5x + 3x \) can be combined by adding coefficients of similar terms, resulting in \( 8x \).
• When variables are involved in expressions under a square root or another radical, factor them when possible to simplify.
• Remember that expressions with fractions and radicals should be approached with techniques like splitting the fraction and expression roots, as seen with \( \frac{b \sqrt{3b}}{2 \sqrt{11}} \).

In our problem, handling both the numerical and variable parts separately allowed for a clearer simplification process, keeping the expression manageable without altering its intent.