Problem 70 - Algebra for College Students

Question

Use the distributive property to help simplify each of the following. All variables represent positive real numbers. \(4 \sqrt{8 n}+3 \sqrt{18 n}-2 \sqrt{72 n}\)

Step-by-Step Solution

Answer

The simplified form is \(5\sqrt{2n}\).

Step 1: Simplify Each Inside the Radical

First, simplify each term by breaking down the number inside the square root into its prime factors. For \( \sqrt{8n} \), express 8 as \( 2^3 \); for \( \sqrt{18n} \), express 18 as \( 2 \times 3^2 \); and for \( \sqrt{72n} \), express 72 as \( 2^3 \times 3^2 \).

Step 2: Apply the Square Root to Factors

Now apply the square root to the factors. For \( \sqrt{8n} \), it becomes \( \sqrt{2^3 \cdot n} = 2\sqrt{2n} \); for \( \sqrt{18n} \), it becomes \( \sqrt{2 \cdot 3^2 \cdot n} = 3\sqrt{2n} \); and for \( \sqrt{72n} \), it becomes \( \sqrt{(2^3)(3^2)n} = 6\sqrt{2n} \).

Step 3: Multiply by Coefficients

Multiply each simplified version by its original coefficient: \(4 \times 2\sqrt{2n} = 8\sqrt{2n}\), \(3 \times 3\sqrt{2n} = 9\sqrt{2n}\), and \( -2 \times 6\sqrt{2n} = -12\sqrt{2n}\).

Step 4: Combine Like Terms

Now, add or subtract the terms since they all simplify to terms involving \( \sqrt{2n} \): \( 8\sqrt{2n} + 9\sqrt{2n} - 12\sqrt{2n} = (8 + 9 - 12) \sqrt{2n}\).

Step 5: Calculate the Final Answer

Perform the arithmetic inside the parentheses: \( (8 + 9 - 12) \sqrt{2n} = 5\sqrt{2n} \). So the simplified form is \( 5\sqrt{2n} \).

Key Concepts

Simplifying Radicals

Simplifying radicals means making a square root expression as simple as possible. This involves finding the prime factors of the number under the square root (radicand) and simplifying those. Consider \( \sqrt{8n} \) as an example. We break down 8 into its prime factors: \( 8 = 2^3 \). This expression can be rewritten as \( \sqrt{2^3 \cdot n} \).

• Take out pairs of prime factors from under the radical sign. A pair of \( 2^2 \) taken out becomes a single 2 in front of the square root.
• This results in \( 2 \sqrt{2n} \).

Breaking an expression into smaller components always helps. It leads to ideas about how to manipulate and simplify, making complex problems more manageable.

Square Roots

Square roots are fundamental in algebra. They tell us what number times itself equals the given number. For our problem, handling radicals like \( \sqrt{8n} \), \( \sqrt{18n} \), and \( \sqrt{72n} \) is key. Each needs its components simplified by applying square root rules.

• For example, \( \sqrt{8n} \) becomes \( 2 \sqrt{2n} \) once simplified.
• Likewise, \( \sqrt{18n} \) leads to \( 3 \sqrt{2n} \), and \( \sqrt{72n} \) simplifies to \( 6 \sqrt{2n} \).

Recognizing and simplifying square roots is crucial for solving such exercises efficiently. Practice makes perfect, so try simplifying different radicals often to build confidence.

Prime Factorization

Prime factorization is breaking down a number into a product of its prime numbers. This is essential when simplifying expressions like square roots. For instance, factorize the number 18 in \( \sqrt{18n} \).

• 18 breaks down as \( 2 \times 3^2 \).
• Similarly, 72 in \( \sqrt{72n} \) turns into \( 2^3 \times 3^2 \).

Understanding prime factorization allows you to simplify radicals effectively. Extract pairs of identical factors from under the radical to simplify expressions. This technique is vital to mastering algebraic simplifications.

Combining Like Terms

Combining like terms refers to summing or subtracting terms with the same variable factor and exponent. In our exercise, once simplified, we have expressions like \( 8\sqrt{2n} \), \( 9\sqrt{2n} \), and \( -12\sqrt{2n} \). These can be combined because they involve the same radical part \( \sqrt{2n} \).

• Add or subtract their coefficients: \( 8 + 9 - 12 \).
• Resulting in: \( 5 \sqrt{2n} \).

Combining like terms helps simplify an expression into its most reduced form. Ensuring terms are alike allows easy calculation and clearer problem solutions, a fundamental algebraic skill.