Problem 82 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt{\frac{72 q^{7}}{25 q^{3}}} $$

Step-by-Step Solution

Answer

\( \frac{6 \sqrt{2} q^2}{5} \)

Step 1: Simplifying Inside the Square Root

We start with the expression under the square root: \( \frac{72 q^{7}}{25 q^{3}} \). First, simplify the fraction by dividing both the numerator and denominator by \( q^3 \), which yields \( \frac{72 q^{4}}{25} \).

Step 2: Separating the Fraction's Square Root

The expression is now inside the square root as \( \sqrt{\frac{72 q^{4}}{25}} \). According to the square root property, this can be separated into \( \frac{\sqrt{72 q^{4}}}{\sqrt{25}} \).

Step 3: Calculating the Denominator's Square Root

Compute the square root of 25. Since \( \sqrt{25} = 5 \), the expression becomes \( \frac{\sqrt{72 q^{4}}}{5} \).

Step 4: Simplifying the Numerator's Square Root

For \( \sqrt{72 q^4} \), note that \( 72 = 36 \times 2 \), and \( \sqrt{36} = 6 \). Similarly, \( \sqrt{q^4} = q^2 \). Thus, \( \sqrt{72 q^4} = \sqrt{36 \times 2} \times q^2 = 6 \sqrt{2} \times q^2 \).

Step 5: Final Expression

Insert the simplified result from Step 4 into the fraction: \( \frac{6 \sqrt{2} \times q^2}{5} \). The expression is fully simplified.

Key Concepts

Algebra Simplification

Algebra simplification is like organizing a messy room. It's about stepping back and finding simpler ways to write expressions without changing their value. Here, the goal is to make operations as straightforward as possible.
To simplify algebraic expressions, one generally:

• Groups like terms together. That means terms that have the same variables and powers.
• Uses properties of operations, such as distributive or associative laws, to combine terms.
• Simplifies the coefficients — that’s the numerical part of the terms.

In the exercise, the original fraction under the square root is first simplified by dividing both parts of the fraction by the same variable power. After that, we used square root properties to further break down the expression, which is exactly what's meant by algebra simplification. By the end of the process, the expression is cleaner and easier to understand, with unnecessary components reduced.

Rational Expressions

Rational expressions are very similar to fractions, but instead of having whole numbers as the numerator and denominator, they can have variables too. They represent ratios of polynomial expressions.
Simplifying rational expressions follows similar rules to simplifying numerical fractions:

• Factorize both the numerator and the denominator wherever possible. This means breaking them down into products of simpler expressions.
• Cancel out like factors that appear in both the numerator and the denominator. This step helps to make the expression as simple as possible.
• Ensure any restrictions on the variables are noted to prevent division by zero.

In the exercise, the rational expression was initially given inside a square root. We first simplified it by using algebraic manipulation—dividing the terms—which is a critical step in making the problem manageable and setting us up for further simplification techniques.

Radical Expressions

Radical expressions involve roots, like square roots or cube roots. These expressions can often be intimidating, but they're just a different way of expressing exponents. For example, \( \sqrt{x} \) is equivalent to \( x^{1/2} \).
When simplifying radical expressions, you can use several rules:

• The product rule: \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). This allows you to break down a single square root into multiple simpler square roots.
• The quotient rule: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). This helps separate complex fractions into more manageable parts.
• Simplifying under the radicals by finding perfect squares or cubes, which makes the expression easier to handle.

In the step-by-step solution, we saw these rules in action. We split the fraction under the radical into two separate square roots, then simplified those individually by factoring. This carefully organized approach makes handling radical expressions much easier and less intimidating.