Problem 113 - Elementary and Intermediate Algebra

Question

The following problems involve addition, subtraction, and multiplication of radical expressions, as well as rationalizing the denominator. Perform the operations and simplify, if possible. All variables represent positive real numbers. $$ \sqrt{\frac{72 m^{8}}{25 m^{3}}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{6m^2\sqrt{2m}}{5} \).

Step 1: Simplify Inside the Square Root

First, simplify the expression inside the square root: \( \frac{72m^8}{25m^3} \). Divide the coefficients and subtract the exponents of like bases. Since 72 and 25 have no common factors, the fraction simplifies to: \[ 72m^{8-3} = 72m^{5}. \]Therefore, the expression is now \( \sqrt{\frac{72m^5}{25}} \).

Step 2: Separate the Fraction Under the Root

Rewrite the square root of a fraction as the fraction of square roots:\[ \sqrt{\frac{72m^5}{25}} = \frac{\sqrt{72m^5}}{\sqrt{25}}.\]

Step 3: Simplify the Denominator

Calculate the square root of the denominator. Since 25 is a perfect square: \[ \sqrt{25} = 5, \]the expression becomes:\[ \frac{\sqrt{72m^5}}{5}. \]

Step 4: Simplify the Numerator

The next step is to simplify \( \sqrt{72m^5} \). - Break down the number 72 into its factors: \( 72 = 36 \times 2 \); since \( 36 \) is a perfect square, \( \sqrt{36} = 6 \).- Thus, \( \sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2} \).- For the variables: \( m^5 = m^4 \times m \), and since \( m^4 \) is a perfect square, \( \sqrt{m^4} = m^2 \) and \( \sqrt{m^5} = m^2\sqrt{m} \).So, \( \sqrt{72m^5} = 6m^2\sqrt{2m} \).

Step 5: Write the Final Expression

Combine the simplified parts of the expression. The expression is now:\[ \frac{6m^2\sqrt{2m}}{5}. \]So, the expression simplifies to \( \frac{6m^2\sqrt{2m}}{5} \).

Key Concepts

Addition and Subtraction of Radicals

When dealing with radicals, addition and subtraction is a bit different than what you might be used to with regular numbers. You can only add or subtract radicals if they have the same index and the same radicand (the number under the radical). In simpler terms, you can think of it like adding or subtracting "like terms" in algebra.

• If you have \( \sqrt{a} + \sqrt{a} \), you can add them to get \( 2\sqrt{a} \).
• But if you have \( \sqrt{a} + \sqrt{b} \), you cannot combine them into one term.

This rule is crucial to remember. Always check that the radicals have the same radicand and index before performing addition or subtraction.
In our original exercise, addition or subtraction wasn't needed, but remember, try simplifying each radical first to check if they become like radicals.

Multiplication of Radical Expressions

Multiplying radicals is generally simpler than adding or subtracting them. The key idea is that you can multiply the numbers inside the radicals together, as long as they have the same index.

• For example, \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \).
• This same rule applies to any index: \( \sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab} \).

You can also multiply the coefficients (the numbers in front of the radicals) separately from the radicals themselves.
If you have \( 3\sqrt{a} \times 2\sqrt{b} \), you multiply the coefficients to get \( 6 \), and the radicals to get \( \sqrt{ab} \). This results in \( 6\sqrt{ab} \).
For the given exercise, multiplication wasn't directly performed, but understanding this rule can help you when dealing with more complex expressions.

Rationalizing the Denominator

Rationalizing the denominator means removing any square roots or radicals from the denominator of a fraction. This step is essential because it's generally preferred to have a rational number (like an integer or a simple fraction) in the denominator.
To rationalize a denominator that is a single square root, you multiply both the numerator and the denominator by the radical present in the denominator.

• For example, if you have \( \frac{1}{\sqrt{b}} \), multiply by \( \frac{\sqrt{b}}{\sqrt{b}} = 1 \) to get \( \frac{\sqrt{b}}{b} \).

In the original problem, we see \( \sqrt{25} = 5 \), which is already a rational number, so no further steps were needed to rationalize.
However, if you encounter a radical denominator not easily simplified, these tips can guide you to ensure your final expression is in its simplest, most acceptable form.