Problem 16 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ 9 \sqrt{\frac{1}{3}} $$

Step-by-Step Solution

Answer

The simplified result of the expression \(9 \sqrt{\frac{1}{3}}\) is approximately 5.193

Step 1: Identify the Values

Identify the values that are included in the problem. Here we have the number 9 and the square root term \( \sqrt{\frac{1}{3}} \).

Step 2: Simplify the Square Root

Simplify the square root. In this case, the square root of 1/3 is approximately \( \sqrt{\frac{1}{3}} = 0.577 \).

Step 3: Multiply the Results

Multiply the simplified square root by the number given outside the root. So, 9 * 0.577 = 5.193.

Key Concepts

Understanding the Square Root

The square root is a mathematical operation that is used to find the original number that was squared to get the given number. For example, the square root of 9 is 3, because 3 squared (3^2) is 9. When simplifying square roots, you'll often deal with irrational numbers, especially when handling fractions or decimals. In this exercise, you're looking at the square root of a rational expression, specifically a fraction. The expression is \( \sqrt{\frac{1}{3}} \), which represents the square root of the fraction \( \frac{1}{3} \). To approximate this value, a calculator or some knowledge of decimal approximations is typically utilized, resulting in \( \sqrt{\frac{1}{3}} \approx 0.577 \). Understanding how square roots work, especially with less common fractions, is a valuable skill for tackling more complex arithmetic and algebraic problems.

Deciphering Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Despite seeming complicated, they can be simplified by following basic rules of fractions and algebra. In the context of simplifying radical expressions, it is essential to be comfortable working with these kinds of fractions, as they often appear under square root signs.

• The given expression \( \frac{1}{3} \) is a simple rational expression.
• When simplifying, always check if the numerator and denominator can be factored further or reduced.
• In our case, \( \frac{1}{3} \) is already in its simplest form.

Handling rational expressions involves recognizing which parts can be reduced or transformed to simplify calculations, especially when square roots are involved.

Mastering the Multiplication of Numbers

Multiplication is one of the four basic operations in arithmetic, crucial to solving many mathematical problems effectively. The operation involves adding a number to itself a certain number of times. When dealing with multiplication in the context of radical expressions or any other expressions, three key considerations can simplify the process:

• Understand the values you're multiplying. Here, it involves 9 and an approximated number \(0.577\), derived from \(\sqrt{\frac{1}{3}}\).
• Use multiplication to combine these values: \(9 \times 0.577 \).
• Calculate the result, which helps you simplify or make sense of the entire expression, resulting in approximately \(5.193\) in this case.

Handling multiplication correctly is integral to getting accurate results when simplifying expressions, especially when involving radicals and irrational numbers.