Problem 333 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{\sqrt{108 p^{5} q^{2}}}{\sqrt{3 p^{3} q^{6}}} $$

Step-by-Step Solution

Answer

The simplified expression is \(2 \frac{p}{q^2}\).

Step 1 - Simplify the Radicals

Simplify both numerator and denominator: \[\frac{\text{Simplify } \sqrt{108 p^{5} q^{2}}}{\text{Simplify } \sqrt{3 p^{3} q^{6}}}\]108 can be factored into \(108 = 2 \times 2 \times 3 \times 3 \times 3\), and 3 can be simplified as \(3 = 3\).

Step 2 - Rewrite the Radicals

Express the simplified terms under the radicals:\[\frac{\text{Simplify } \sqrt{2^2 \times 3^3 \times p^5 \times q^2}}{\text{Simplify } \sqrt{3 \times p^3 \times q^6}}\]

Step 3 - Simplify Each Radical

Simplify the square roots in both the numerator and the denominator:\[\frac{6 p^{2.5} q}{\text{Simplify } 3 p^{1.5} q^3}\]

Step 4 - Divide the Terms

Divide both numerator and denominator by their common factors:\[\frac{\text{Simplify }6 \times p^{2.5} \times q}{\text{Simplify }3 \times p^{1.5} \times q^3}= \frac{6}{3} \times \frac{p^{2.5}}{p^{1.5}} \times \frac{q}{q^3} = 2 p q^{-2}\]

Step 5 - Simplify the Expression

Combine the terms with the same base, finally yielding:\[2 p q^{-2} = 2 \frac{p}{q^2}\]

Key Concepts

simplifying algebraic expressions

When simplifying algebraic expressions, your main goal is to combine like terms and reduce the expression to its simplest form.
Identify like terms: Look for terms that have the same variables raised to the same power.
Combine them by adding or subtracting their coefficients.
Always keep the variable part intact.

For multiplying and dividing, use the laws of exponents.
For example, when dividing like bases, subtract the exponents.
When multiplying, add the exponents.
Simplify wherever possible to get the expression in its simplest form.
An example follows below:

\[2 \times 3x^2 - 5x + 7 + 2x^2 = 5x^2 - 5x + 9\].

radicals and roots

Radicals and roots involve expressions with square roots and higher-order roots.
For example, \(\root{2}{x}\) or \(\root{3}{y}\).
To simplify radicals:

• Factor the number or expression under the radical into its prime factors.
• Rewrite the radical in terms of these factors.
• Look for pairs of factors (for square roots), groups of three (for cube roots), etc.
• Simplify by moving those pairs/groups outside the radical.

Here’s a short example:
\(\root{2}{108} = \root{2}{2^2 \times 3^3} = 6 \root{2}{3}\).
Understanding how to simplify roots will help in performing operations involving radicals like addition, subtraction, multiplication, and division.

fraction simplification

Simplifying fractions means reducing them to their smallest form.
Here’s how you can do this:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.

For algebraic fractions, you need to simplify both the numeric and the variable parts.
Cancel out common factors in the numerator and denominator.

Here’s an example:
\[ \frac{36 p^3}{12 p} \].
Find the GCD of 36 and 12, which is 12.
Divide both by 12: \[ \frac{36 p^3}{12 p} = \frac{3 p^3}{p} = 3 p^2 \].
The fraction is now simplified.

algebraic division

Algebraic division involves dividing algebraic expressions.
In simple cases, it is much like fraction division but includes variables.
The critical step here is to use the laws of exponents:

• When dividing like bases, subtract the exponents.

Consider this example:
\( \frac{p^5}{p^3} \).
Apply the law of exponents: \( p^{5-3} = p^2 \).
For more complicated expressions, ensure you simplify each part before dividing.
Let’s see a full expression:
\( \frac{2p^5q^2}{pq^3} \).
Divide both terms separately: \( 2 \times \frac{p^5}{p} \times \frac{q^2}{q^3} \).
This becomes: \( 2 \times p^4 \times q^{-1} = 2 p^4 \frac{1}{q} \) or \( \frac{2 p^4}{q} \).
There you have it, simplified!