Problem 26 - Algebra for College Students

Question

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt{80 x y^{2}}\)

Step-by-Step Solution

Answer

The simplest radical form is \(4y\sqrt{5x}\).

Step 1: Prime Factorization

First, we perform the prime factorization of the number inside the square root. The number 80 can be broken down into its prime factors: \[ 80 = 2^4 imes 5^1 \] Thus, inside the square root, we have: \[ \sqrt{80x y^2} = \sqrt{2^4 \cdot 5 \cdot x \cdot y^2} \]

Step 2: Apply the Square Root

Next, we can apply the square root to each factor separately. Remember that the square root of a product is the product of the square roots, and that \(\sqrt{a^2} = a\): \[ \sqrt{2^4 \cdot 5 \cdot x \cdot y^2} = \sqrt{(2^2)^2} \times \sqrt{5} \times \sqrt{x} \times \sqrt{y^2} \] Simplifying, we get: \[ 2^2 \times y \times \sqrt{5x} \]

Step 3: Simplify the Expression

Now, simplify the expression by multiplying any integer coefficients and simplifying under the square root: \[ 2^2 = 4 \]Thus, the expression simplifies to: \[ 4y \sqrt{5x} \]

Step 4: Conclusion

After simplifying the radical expression, the simplest radical form of \(\sqrt{80xy^2}\) is \(4y\sqrt{5x}\).

Key Concepts

Prime Factorization

Prime factorization is a fundamental concept in simplifying radical expressions. It's the process of breaking down a number into its basic building blocks—prime numbers. Prime numbers are integers greater than 1 that have no divisors other than 1 and themselves, such as 2, 3, 5, 7, etc. When you have a composite number like 80, you can express it as a product of prime numbers.

For example, the number 80 can be decomposed as follows:

• 80 is even, so it’s divisible by 2: 80 divided by 2 gives 40.
• 40 is also even, so divide by 2 again: 40 divided by 2 gives 20.
• 20 divided by 2 again gives 10.
• 10 divided by 2 gives 5.

At this point, we reach 5, which is a prime number. So, 80 can be expressed in terms of prime factors as \(2^4 \times 5^1\). Prime factorization is essential to simplifying radicals, as it helps identify squares (pairs of identical factors), which can further be simplified.

Square Root Properties

The properties of square roots are essential when simplifying radical expressions. Understanding how to handle square roots properly is a key skill. Here are some basic square root properties to keep in mind:

• The square root of a product is the product of the square roots: \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \).
• The square root of a squared term gives the original term: \( \sqrt{a^2} = a \).

Using these properties, you can simplify square roots effectively. Take the expression \( \sqrt{80xy^2} \). Based on our prime factorization, we have:

• \( \sqrt{2^4 \cdot 5 \cdot x \cdot y^2} \).
• The square roots are applied separately: \( \sqrt{(2^2)^2} \times \sqrt{5} \times \sqrt{x} \times \sqrt{y^2} \).
• \( \sqrt{(2^2)^2} = 2^2 = 4 \), and \( \sqrt{y^2} = y \).

By identifying and simplifying these terms, you end up simplifying the overall expression, thus giving you the simplest radical form.

Algebraic Expressions

Algebraic expressions involving radicals can look complex, but they follow the same foundational principles as any other algebraic expression. When simplifying, it’s important to manage both the numeric and variable components inside the square root.

To simplify an algebraic radical expression like \( \sqrt{80xy^2} \):

• First, perform prime factorization (already discussed) which helps simplify the numeric part.
• Apply square root properties to break down and simplify both the numeric and variable parts.
• Combine results with constants and like terms outside the radical for the simplest expression.

With our example, \( \sqrt{80xy^2} \) simplifies to \( 4y\sqrt{5x} \), combining the constant 4 from \( 2^2 \) and the variable y from \( y^2 \). The goal is always to represent the expression in its simplest form by reducing as many terms as possible using algebraic principles.