Problem 77 - Elementary Algebra

Question

Simplify. $$ \sqrt{98 m^{5}} $$

Step-by-Step Solution

Answer

7m^2 \sqrt{2m}

Step 1: Prime Factorization

First, perform prime factorization of 98. The factors of 98 are 2 and 49, and 49 can be broken down further into 7 and 7. Thus:\[ 98 = 2 \times 7^2 \]

Step 2: Separate Inside the Square Root

Rewrite the expression under the square root using the results from prime factorization and the exponent for the variable:\[ \ \sqrt{98 m^{5}} = \sqrt{2 \times 7^2 \times m^{5}} \]

Step 3: Apply Square Root to Each Factor

Apply the square root to each factor separately. Remember that:\( \ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \):\[ \ \sqrt{2 \times 7^2 \times m^{5}} = \sqrt{2} \times \sqrt{7^2} \times \sqrt{m^{5}} \]

Step 4: Simplify Each Term

Simplify \(\sqrt{7^2}\) and \(\sqrt{m^5}\):\[ \ \sqrt{7^2} = 7 \quad \text{and} \quad \sqrt{m^{5}} = \sqrt{(m^2)^2 \times m} = m^2\sqrt{m} \]So the expression becomes:\[ \ \sqrt{2} \times 7 \times m^2 \sqrt{m} \]

Step 5: Combine like Terms

Combine the simplified terms:\[ \ 7m^2 \sqrt{2m}\]

Key Concepts

prime factorization

Prime factorization is the process of breaking down a number into its prime components. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For example, the number 98 can be factorized into 2 and 49. We can further break down 49 into 7 and 7, thus the prime factors of 98 are:

• 2
• 7
• 7

So, 98 can be written as:
\[ 98 = 2 \times 7^2 \]This step is crucial in simplifying square roots, as it allows us to break down the number into smaller, more manageable parts.

square root properties

The properties of square roots help in simplifying expressions under the radical symbol. One of the most important properties is that the square root of a product is the product of the square roots:

• \[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \]

For example, in the given problem, we separate the factors inside the square root:
\[ \sqrt{98 m^{5}} = \sqrt{2 \times 7^2 \times m^{5}} \]Then, apply the square root property to each factor separately to get:
\[ \sqrt{2 \times 7^2 \times m^{5}} = \sqrt{2} \times \sqrt{7^2} \times \sqrt{m^{5}} \]Understanding this property is key to further simplifying radical expressions.

simplifying algebraic expressions

Simplifying algebraic expressions involves combining like terms and reducing expressions to their simplest form. In the context of the problem:

• Simplify the square root of a perfect square:
  • \[ \sqrt{7^2} = 7 \]
• Handle variables with exponents under the square root:
  • \[ \sqrt{m^{5}} = \sqrt{(m^2)^2 \times m} = m^2 \sqrt{m} \]

By applying these steps, the expression simplifies to a more manageable form:
\[ \sqrt{2} \times 7 \times m^2 \sqrt{m} \]Combining like terms gives the final simplified expression:
\[ 7m^2 \sqrt{2m} \]This process aids in making complex algebraic expressions easier to work with.

radical expressions

Radical expressions contain a radical symbol, usually a square root. They often require simplification to make them easier to handle in equations and other mathematical operations. The problem at hand is a good example:
The expression \( \sqrt{98 m^{5}} \) can be rewritten using the results from prime factorization and properties of square roots. This allows us to break down the expression step-by-step and simplify each part individually:
Simplifying radical expressions is important in algebra and is frequently encountered in various areas of mathematics, including solving equations and working with geometric concepts.
Understanding how to manipulate these expressions is crucial for simplifying and solving more complex problems.