Problem 51 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{m^{6} n^{8} p^{12} q^{20}} $$

Step-by-Step Solution

Answer

Question: Simplify the radical expression: √(m^6 * n^8 * p^12 * q^20) Answer: m^3 * n^4 * p^6 * q^10

Step 1: Apply the multiplication rule for square roots

The multiplication rule for square roots states that the square root of a product equals the product of the square roots of each factor. So, we can rewrite the expression as follows: $$ \sqrt{m^{6}n^{8}p^{12}q^{20}} = \sqrt{m^{6}}\sqrt{n^{8}}\sqrt{p^{12}}\sqrt{q^{20}} $$

Step 2: Apply the power rule for square roots

The power rule for square roots states that the square root of a number raised to a certain power equals that number raised to half the original power. Applying this rule to our expression, we get: $$ \sqrt{m^{6}}\sqrt{n^{8}}\sqrt{p^{12}}\sqrt{q^{20}} = m^{(6/2)}n^{(8/2)}p^{(12/2)}q^{(20/2)} $$

Step 3: Perform calculations

Now, we just need to simplify the fractions, resulting in the following expression: $$ m^{(6/2)}n^{(8/2)}p^{(12/2)}q^{(20/2)} = m^{3}n^{4}p^{6}q^{10} $$

Step 4: Write the final answer

After simplifying the radical expression, we get the final answer: $$ \sqrt{m^{6}n^{8}p^{12}q^{20}} = m^{3}n^{4}p^{6}q^{10} $$

Key Concepts

Multiplication Rule for Square Roots

Understanding the multiplication rule for square roots is key to simplifying radical expressions. It establishes that the square root of a product of numbers can be broken down into the product of the square roots of each individual number. For example:
\[\begin{equation}\sqrt{ab} = \sqrt{a} \times \sqrt{b}\end{equation}\]
This is particularly helpful because it allows us to simplify complex expressions piece by piece, rather than all at once. By distributing the radical across the multiplication, it can make complex problems more manageable. Remember, this rule only applies when all the numbers involved are non-negative since the square root of a negative number involves imaginary numbers, which have a different set of rules.

Power Rule for Square Roots

The power rule for square roots bridges the gap between radical expressions and exponents. The rule says that for any positive number 'a' and even exponent '2n':
\[\begin{equation}\sqrt{a^{2n}} = a^n\end{equation}\]
This rule essentially halves the exponent. It’s like unpacking the square root by dividing the exponent by two, which makes sense since taking the square root is the inverse operation of squaring a number. In practice, when you encounter a radical with a power inside, such as \(\sqrt{m^{6}}\), you can simplify it to \(m^{3}\) because \(6/2 = 3\). This will drastically simplify the expression and is particularly useful when dealing with higher powers inside radicals.

Radical Simplification

Radical simplification involves reducing a radical expression to its simplest form. This process usually entails identifying and pulling out powers of numbers that are hidden within the radical.
For instance, in the provided exercise, \(\sqrt{m^{6}n^{8}p^{12}q^{20}}\) is broken down into separate radicals before simplifying each part. When practicing radical simplification, it’s important to:

• Look for perfect squares or other powers that match the radical's index.
• Break down the expression into smaller parts if applicable.
• Use the power and multiplication rules for square roots.
• Remember to simplify all coefficients and terms both inside and outside the radical.

By systematically applying these strategies, you can simplify any radical expression efficiently.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain ordinary numbers, variables (like \(n\), \(m\), and others), and the operations between them. When simplifying algebraic expressions, especially those containing radicals, it's crucial to:

• Combine like terms, which are terms with the same variables and exponents.
• Utilize distributive properties to simplify expressions.
• Apply the relevant exponent rules such as the power rule.

Understanding how to manipulate these expressions is essential for algebraic proficiency. In the case of our exercise, we took the algebraic expression inside the square root and applied exponent rules to transform it into an equivalent expression without radicals, which is generally the form desired for a simplified solution.