Problem 24 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{m^{7}} $$

Step-by-Step Solution

Answer

Question: Simplify the radical expression \(\sqrt{m^7}\). Answer: \(\sqrt{m^7} = m^3 \cdot \sqrt{m}\)

Step 1: Identify the largest even power of the expression within the root

In this case, we have \(m^7\) inside the square root. The largest even power within m^7 is \(m^6\).

Step 2: Apply the square root rule

The square root of m to the power of 2n is m to the power of n. In this case, we have \(m^6\), and we will apply the rule as follows: $$ \sqrt{m^6} = m^3 $$

Step 3: Rewrite the radical expression

Now, rewrite the initial expression using the result from step 2: $$ \sqrt{m^7} = \sqrt{m^6 \cdot m} $$

Step 4: Distribute the square root

Next, distribute the square root over the multiplication inside the root. This can be done because of the property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\): $$ \sqrt{m^6 \cdot m} = \sqrt{m^6} \cdot \sqrt{m} $$

Step 5: Substitute the simplified term

Now, substitute the simplified square root \(\sqrt{m^6} = m^3\) from step 2 back into the expression: $$ m^3 \cdot \sqrt{m} $$

Step 6: Present the simplified expression

Finally, we have the simplified expression for the given radical: $$ \sqrt{m^7} = m^3 \cdot \sqrt{m} $$

Key Concepts

Square Root Rules

Understanding square root rules is fundamental when simplifying radical expressions. A square root essentially asks the question: 'Which number, when multiplied by itself, gives the original number under the root?'. For example, the square root of 9, denoted as \( \sqrt{9} \), is 3 because \( 3^2 = 9 \).

Key rules for square roots include:

• \( \sqrt{a^2} = a \) (assuming \( a \) is a non-negative number).
• \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \) which is known as the product property.
• For any positive whole number \( n \) and non-negative \( a \) and \( b \) when the square root of a fraction is taken, it distributes across the numerator and the denominator: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) if \( b eq 0 \) .

These rules are the building blocks for simplifying radical expressions and can solve a variety of problems when applied appropriately.

Radical Simplification Steps

When simplifying radical expressions involving square roots, there are strategic steps one can follow for a systematic solution.

Firstly, identify the largest even power within the radicand. This makes it easier to extract square roots since even powers have straightforward square roots. Then, apply the square root to the even power separately. Rewrite the expression to separate the terms inside the radical if necessary, and distribute the square root over the multiplication inside the root. Finally, substitute the simplified term back into the expression and combine any like terms.

By applying these steps in order, simplification becomes more manageable and less prone to errors, which enhances understanding and improves problem-solving efficiency.

Even Power Extraction

Even power extraction plays a critical role in the simplification of radical expressions. The 'even power' here refers to expressions like \( n^{2k} \) where \( n \) is the base and \( 2k \) denotes an even exponent. When extracting an even power from under a square root, the process simplifies significantly because square roots and even powers are inherently compatible.

For instance, the square root of \( n^{2k} \) is simply \( n^k \). This rule allows us to rewrite expressions by taking out the highest even power from the radicand. In the context of our exercise, \( m^7 \) is rewritten by extracting \( m^6 \) as it is the highest even power of \( m \) which simplifies to \( m^3 \) when the square root is applied. This process of even power extraction clarifies the simplification process and minimizes the complexity of the radicals we are working with.

Properties of Square Roots

The properties of square roots are critical to correctly simplifying radical expressions. These properties are mathematical truths that allow for the manipulation and simplification of square roots.

They include the concept of perfect squares, the multiplicative and division properties for roots, and the idea that a square root of a square neutralizes the radical, resulting in the absolute value of the base. A fundamental property is that for any nonnegative number \( a \) and \( b \) if \( a = b^2 \) then \( \sqrt{a} = b \). Additionally, \( \sqrt{a^2} = |a| \) since square roots yield non-negative results, emphasizing the absolute value.

These properties ensure that the simplification of square roots isn't just a series of arbitrary steps, but a logical process that follows specific, consistent rules.