Problem 68 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{\frac{72 x^{2} y^{3}}{5}} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the given radical expression is: \(6x \sqrt{\frac{2y^3}{5}} \).

Step 1: Break Down The Numerator And Denominator Into Prime Numbers

Since we are dealing with a square root, we need to look for pairs of the same number inside the expression. First, let's find the prime factorization of the numbers: For the numerator (inside the square root): $$ 72 x^{2} y^{3} = 2^3 \cdot 3^2 \cdot x^2 \cdot y^3 $$ For the denominator (inside the square root): $$ 5 = 5^1 $$ Now, let's put this into our expression: $$ \sqrt{\frac{2^3 \cdot 3^2 \cdot x^2 \cdot y^3}{5^1}} $$

Step 2: Simplify The Expression By Taking Out Square Roots

Since it's a square root, we can only take out numbers if they appear twice in the expression (i.e., in pairs). We do not have any pairs of 5's, which means we cannot simplify the denominator further. However, we can take out pairs from the numerator: - One pair of 2's (i.e., \(2^2\)); - One pair of 3's (i.e., \(3^2\)); - One pair of x's (i.e., \(x^2\)); So far, our expression is: $$ 2 \cdot 3 \cdot x \cdot \sqrt{\frac{2 \cdot y^3}{5}} $$

Step 3: Rewrite The Simplified Expression In A More Compact And Readable Form

After completing the simplification process, let's rewrite the expression to make it more compact and easier to read: $$ 6x \sqrt{\frac{2y^3}{5}} $$ So, the simplified form of the given radical expression is: $$ \sqrt{\frac{72 x^{2} y^{3}}{5}} = 6x \sqrt{\frac{2y^3}{5}} $$

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its simplest building blocks, which are prime numbers. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. Understanding prime factorization is crucial when working with radical expressions because it allows us to identify and group terms into pairs, making it possible to extract square roots.To perform prime factorization, we start with any composite number and repeatedly divide it by the smallest prime until we're left with prime numbers.
For example, the number 72 can be broken down as follows:

• 72 ÷ 2 = 36
• 36 ÷ 2 = 18
• 18 ÷ 2 = 9
• 9 ÷ 3 = 3
• 3 ÷ 3 = 1

Thus, 72 is expressed as the product of primes: \(2^3 \cdot 3^2\). In algebraic contexts, you can also do this with variables by expressing their exponents as sums of 1 to distinguish pairs.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because \(3 \times 3 = 9\). Simplifying a square root involves removing paired factors from under the radical sign and placing them in front of it.When dealing with square roots of algebraic expressions, like \(\sqrt{\frac{72 x^{2} y^{3}}{5}}\), the goal is to simplify the expression by pulling out even powers from under the square root:

• If a term is squared, it can move outside the radical as its base. For example, \(x^2\) becomes \(x\).
• If a term is not squared, it remains under the square root sign, like \(y^3\), which keeps one "y" under the square root since it cannot be completely paired.

Understanding and manipulating square roots is vital for simplifying complex algebraic expressions and solving equations in mathematics.

Algebraic Expressions

An algebraic expression includes numbers, variables, and operations in a meaningful arrangement. These expressions can represent real-world values when you substitute numbers for variables. In the example given, \(72 x^{2} y^{3}\) is an algebraic expression. It combines numerical coefficients (72) with variables raised to specific powers (such as \(x^2\) and \(y^3\)).Simplification of algebraic expressions often involves reducing these expressions to their simplest form through factoring and collecting like terms. Algebraic expressions under a square root, like in our problem, require special techniques:

• Identify and factor numerals, often using prime factorization.
• Group and simplify variables using their exponents to prepare for extracting roots.

Through understanding algebraic expressions, students can enhance their ability to manipulate and solve equations, which is key in higher-level math.

Rational Expressions

A rational expression is similar to a fraction, where both the numerator and the denominator are polynomials. Simplifying rational expressions involves factoring numerators and denominators before cancelling out common factors.In the context of the problem, the expression \(\sqrt{\frac{72 x^{2} y^{3}}{5}}\) is a radical expression with a rational component. Here, we simplify it by identifying common factors – both numerical and algebraic – and using techniques like prime factorization and simplifying the square root.A key step in solving rational expressions with radicals involves ensuring there's no radical in the denominator whenever possible, and expressing the solution in its simplest form. In mathematical practice, this often means performing operations to reduce complexity and improve comprehension.