Problem 139 - Elementary Algebra

Question

Simplify. $$ \sqrt{\frac{27 p^{2} q}{108 p^{5} q^{3}}} $$

Step-by-Step Solution

Answer

\( \frac{1}{2 p^{3/2} q} \)

Step 1 - Simplify Under the Square Root

First, simplify the expression inside the square root by dividing the numerator by the denominator. Simplify the constants and variables separately.

Step 2 - Simplify the Constant

Divide the constant terms 27 and 108: \ \ \[\frac{27}{108} = \frac{1}{4} \ \ \]

Step 3 - Simplify the Variables

Simplify the variable terms: \ \ \[\frac{p^{2} q}{p^{5} q^{3}} = \frac{p^{2}}{p^{5}} \cdot \frac{q}{q^{3}} = p^{2-5} \cdot q^{1-3} = p^{-3} \cdot q^{-2} = \frac{1}{p^{3} q^{2}} \ \ \]

Step 4 - Combine Results

Combine the simplified constant and variable terms: \ \ \[ \sqrt{\frac{1}{4 p^3 q^2}} = \sqrt{\frac{1}{4}} \cdot \sqrt{\frac{1}{p^3 q^2}} \ \ \]

Step 5 - Simplify the Square Root

Simplify the square roots separately: \ \ \[\sqrt{\frac{1}{4}} = \frac{1}{2} \ \ \sqrt{\frac{1}{p^3 q^2}} = \frac{1}{p^{3/2} q}\ \ \]

Step 6 - Combine Final Results

Combine the results from Step 5: \ \ \[ \frac{1}{2} \cdot \frac{1}{p^{3/2} q} = \frac{1}{2 p^{3/2} q} \ \ \]

Key Concepts

Square Roots

Square roots represent a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 * 3 equals 9.
When dealing with square roots in algebra, especially those including variables, simplifying them can make the expressions much easier to understand and work with.
In our problem, we have a square root that contains a fractional expression inside it. To simplify a square root of a fraction, you take the square root of the numerator and the square root of the denominator separately.
For instance: \[ \frac{a}{b} = \frac{\sqrt{a}}{\sqrt{b}} \] where 'a' and 'b' are any algebraic expressions.

Variable Simplification

Simplifying variables involves reducing the expression to its simplest form, often by canceling out terms. When you see variables in both the numerator and denominator, you can simplify by dividing out common variables.
In the given problem, we need to simplify the variable terms p and q inside the square root.

• First look at each variable separately.
• For p, you have p^2 over p^5, which simplifies to p^{2-5} = p^{-3}.
• For q, you have q^{1} over q^{3}, which simplifies to q^{1-3} = q^{-2}.

This turns our fraction into \[ \frac{1}{p^3 q^2} \] while simplified.

Fractional Exponents

Fractional exponents are another way to express roots. For example, the square root of x can be written as x^{1/2}. This is particularly useful when dealing with variable expressions because it allows us to apply exponent rules for further simplification.
Our expression \[ \frac{1}{p^3 q^2} \] can thus be handled by integrating the fractional exponents:
We rewrite the expression inside the square root as \[ \frac{1}{4 \times p^3 q^2} \] and then apply the square root as \[ \sqrt{\frac{1}{4 p^3 q^2}} = \sqrt{\frac{1}{4}} \times \sqrt{\frac{1}{p^3 q^2}} \]
Each part of this expression can be rewritten using fractional exponents such as \[ \frac{1}{p^3} = p^{-3/2} \] and finally combined.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator contain algebraic expressions. Simplifying rational expressions often means factoring and finding a common ground where terms can be canceled out.
In the given problem, we started with a rational expression inside the square root:

• First, simplify the constants: \frac{27}{108} = \frac{1}{4}
• Next, deal with the variables as discussed previously.
• Finally, apply the square root to both the simplified numerator and denominator separately.

This led us to the final simplified form: \[ \frac{1}{2 p^{3/2} q} \]