Problem 56 - College Algebra

Question

For the following exercises, simplify each expression. $$ \sqrt{\frac{42 q}{36 q^{3}}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{\sqrt{7}}{q\sqrt{6}} \).

Step 1: Simplify the fraction under the square root

First, observe the expression inside the square root: \( \frac{42q}{36q^3} \). Simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 6.Divide the numbers:\[ \frac{42}{36} = \frac{42 \div 6}{36 \div 6} = \frac{7}{6} \]Now the expression is \( \frac{7q}{6q^3} \).

Step 2: Simplify the algebraic expression

Now, simplify the expression \( \frac{7q}{6q^3} \) further by reducing the powers of \( q \).Subtract the exponent of \( q \) in the numerator from the exponent of \( q \) in the denominator:\[ q^{1 - 3} = q^{-2} \]Therefore, the simplified expression is \( \frac{7}{6q^2} \).

Step 3: Apply the square root

Apply the square root to the simplified expression:\[ \sqrt{\frac{7}{6q^2}} \]This is the same as:\[ \frac{\sqrt{7}}{\sqrt{6q^2}} \]Further simplify by finding the square root of the denominator:\[ \sqrt{6q^2} = \sqrt{6} \times q \] since \( q^2 \) has a square root of \( q \).

Step 4: Simplify the final expression

Now, the expression becomes:\[ \frac{\sqrt{7}}{q\sqrt{6}} \]This is the simplest form of the expression \( \sqrt{\frac{42q}{36q^3}} \).

Key Concepts

Square Root Simplification

The process of square root simplification involves taking the square root of a fraction and simplifying it as much as possible. When dealing with square roots, it's important to separate the square root of the numerator and the denominator. For example, when you have an expression like \( \sqrt{\frac{7}{6q^2}} \), you can rewrite it as \( \frac{\sqrt{7}}{\sqrt{6q^2}} \).

In cases where the denominator or numerator includes a squared term, such as \( q^2 \), you can simplify it to \( q \) because the square root of \( q^2 \) is \( q \). This makes the expression easier to manage and understand. After breaking down the components, ensure there are no perfect squares left under the square root, simplifying it as far as possible to reach the simplest form.

Greatest Common Divisor

Finding the greatest common divisor (GCD) is essential when simplifying fractions. The GCD of two numbers is the largest number that can divide both without leaving any remainder. For a fraction \( \frac{42q}{36q^3} \), the numerical GCD of 42 and 36 is 6.

Here's how you calculate that:

• List the factors of each number.
• For 42, the factors are 1, 2, 3, 6, 7, 14, 21, and 42.
• For 36, they are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

The largest common factor is 6. Using the GCD, divide both the numerator and the denominator to simplify the fraction.

Fraction Simplification

Simplifying fractions involves both numerical and algebraic expressions. First, use the GCD to simplify numbers, then apply rules for algebraic terms. In the expression \( \frac{7q}{6q^3} \), you simplify by addressing both the coefficients and the variables.

Once the numerical part is reduced using the GCD, focus on the variables. If both numerator and denominator have the same variable, subtract the exponents. For example, subtract 3 from 1 in \( q^3 \) and \( q^1 \), resulting in \( q^{-2} \) in the denominator. The final fraction will have a simpler form \( \frac{7}{6q^2} \), with the variable neatly managed.

Exponent Rules

Exponent rules are crucial in simplifying algebraic expressions involved in fractions and radical simplification. For instance, when dealing with the expression \( \frac{7q}{6q^3} \), apply the rule of subtracting exponents. The expression becomes \( q^{1-3} = q^{-2} \), a crucial step in obtaining a simpler form.

These rules dictate how you handle powers when multiplying, dividing, and changing variable expressions:

• Multiplying powers with the same base involves adding exponents.
• Dividing powers with the same base involves subtracting exponents.
• Raising a power to another power multiplies the exponents.

Mastering these rules will ease the handling of complex expressions, aiding in finding the simplest form as required in algebraic simplification.