Problem 56 - College Algebra

Question

Simplify each expression. $$\sqrt{\frac{42 q}{36 q^{3}}}$$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Fraction Inside the Square Root

First, look at the fraction \( \frac{42q}{36q^3} \). Simplify it by dividing both the numerator and the denominator by the greatest common factor (GCF) of the numbers, which is 6. This gives us \( \frac{7q}{6q^3} \).

Step 2: Simplify the Variables

Next, simplify the variable part of the fraction. In the expression \( \frac{7q}{6q^3} \), we can cancel \( q \) in the numerator with one \( q \) out of the three \( q \)s in the denominator. This results in \( \frac{7}{6q^2} \).

Step 3: Apply the Square Root to the Fraction

Now, apply the square root to the simplified fraction \( \sqrt{\frac{7}{6q^2}} \). This can be expressed as \( \frac{\sqrt{7}}{\sqrt{6q^2}} \).

Step 4: Simplify the Denominator

Since \( \sqrt{6q^2} = \sqrt{6} \cdot \sqrt{q^2} \), and \( \sqrt{q^2} = q \), simplify it to \( \sqrt{6}q \).

Step 5: Write the Final Expression

Combine everything to get the final simplified expression as \( \frac{\sqrt{7}}{\sqrt{6}q} \).

Key Concepts

Factoring Algebraic Expressions

Factoring is the process of breaking down an expression into simpler components called factors. It is an essential skill in algebra as it simplifies computations and helps solve equations.
To factor algebraic expressions, identify common elements in terms. Often, you'll look for the greatest common factor (GCF) among coefficients and associated variable parts.

• Factoring can transform expressions into more manageable forms.
• It is often used to simplify fractions by canceling out common factors in numerators and denominators.

In our example, we identified that the GCF of the numbers 42 and 36 is 6. Factoring these numbers by 6 makes simplification straightforward, reducing the initial fraction to \( \frac{7q}{6q^3} \). This initial simplification is critical for further manipulation.

Variables Manipulation

Variables represent unknown or changeable values in algebraic expressions and equations. Manipulating variables means performing operations like addition, subtraction, multiplication, division, and more.
In simplification, particularly in fractions, it involves canceling or combining like terms.

• Ensure you perform operations on both sides of an equation or both parts of a fraction to maintain equality.
• When manipulating variables, be mindful of the exponent rules.

In the example \( \frac{7q}{6q^3} \), we simplified by canceling one q from the numerator and denominator. Understanding how to handle variables and their exponents is key to continuing the simplification process.

Square Root Simplification

The square root is a function that reverses squaring a number. Simplifying expressions involving square roots helps in solving equations and reducing expression complexity.
It often demands breaking down numbers or factors under the square root to their simplest form.

• Apply the property \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \) to separate or combine terms.
• Look for perfect squares under the root for simpler expressions.

In our example, we used the property to handle the square root of a fraction: \( \sqrt{\frac{7}{6q^2}} \) was split into \( \frac{\sqrt{7}}{\sqrt{6q^2}} \). Then, simplifying \( \sqrt{q^2} \) to q, refined our expression further, resulting in a neatly simplified form.

Greatest Common Factor (GCF)

The greatest common factor is the largest number that divides two or more numbers without leaving a remainder. Identifying the GCF is critical in simplifying expressions, particularly fractions, by reducing them to their lowest terms.
Finding the GCF involves listing factors of numbers or using prime factorizations.

• GCF is vital when simplifying fractions, ensuring numbers are broken down properly.
• It helps in recognizing common factors in more complex expressions.

For the given expression, we identified the GCF of 42 and 36 as 6. By dividing both the numerator and the denominator by this GCF, we significantly reduced the complexity, setting up for further simplification steps.