Problem 59 - Intermediate Algebra

Question

Simplify by combining like radicals. $$ \sqrt{98}-\sqrt{50}-\sqrt{72} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-4\sqrt{2}\).

Step 1: Prime Factorize the Radicands

Begin by breaking down the numbers under each square root into their prime factors. 1. For \(\sqrt{98}\), the prime factorization is \(98 = 2 \times 7^2\).2. For \(\sqrt{50}\), the prime factorization is \(50 = 2 \times 5^2\).3. For \(\sqrt{72}\), the prime factorization is \(72 = 2^3 \times 3^2\).

Step 2: Simplify the Radicals

Use the prime factorizations to simplify each radical.1. \(\sqrt{98} = \sqrt{2 \times 7^2} = 7\sqrt{2}\).2. \(\sqrt{50} = \sqrt{2 \times 5^2} = 5\sqrt{2}\).3. \(\sqrt{72} = \sqrt{2^3 \times 3^2} = \sqrt{8 \times 9} = 3\sqrt{8} = 3\times 2\sqrt{2} = 6\sqrt{2}\).

Step 3: Combine Like Terms

Combine the simplified radicals with like terms, which all include \(\sqrt{2}\).The expression \(7\sqrt{2} - 5\sqrt{2} - 6\sqrt{2}\) simplifies as follows:- First, combine the coefficients of \(\sqrt{2}\): \(7 - 5 - 6 = -4\).Thus, the expression simplifies to \(-4\sqrt{2}\).

Key Concepts

Prime Factorization

Prime factorization is a method used to express any number as a product of prime numbers, which are numbers greater than 1 that cannot be divided evenly by any other numbers except 1 and themselves. This is helpful in simplifying radicals, like square roots, because it lets us see how numbers break down into simpler components.

• To begin prime factorization, choose a number you want to break down. For example, let's take the number 98.
• Divide it by the smallest prime number (in this case 2), and then keep dividing the resulting number by its smallest possible prime.
• Continue until all resulting factors are prime numbers. So, 98 can be broken down into 2 and 7², giving us the prime factorization: 2 × 7².

Prime factorization makes it easy to simplify radicals, because once we express numbers under a square root as a product of primes, we can easily identify and pull out perfect squares. This is how \(\sqrt{98}\) can be simplified to \(7\sqrt{2}\). By following similar steps with numbers like 50 and 72, we identify what can be taken out from under the square root.

Like Terms

Understanding and working with like terms is essential when simplifying expressions, especially in algebra. Like terms are terms in an expression that have the same variable raised to the same power. In the case of simplifying radicals, like terms have the same radical part.

• In the case of our exercise, we focus on simplifying radicals like \(\sqrt{98}\), \(\sqrt{50}\), and \(\sqrt{72}\). After simplification, they become \(7\sqrt{2}\), \(5\sqrt{2}\), and \(6\sqrt{2}\).
• As seen, they all have the radical part \(\sqrt{2}\), making them like terms.

When combining like terms in the expression \(7\sqrt{2} - 5\sqrt{2} - 6\sqrt{2}\), you only need to add or subtract the coefficient (the number in front of the radical) of the same radical part. So, you simply calculate: \(7\) (from \(7\sqrt{2}\)) minus \(5\) (from \(5\sqrt{2}\)) minus \(6\) (from \(6\sqrt{2}\)). This results in \(-4\), and we get the simplified expression as \(-4\sqrt{2}\). This understanding makes it easier to combine terms properly and simplify expressions.

Square Roots

Square roots are a fundamental concept in mathematics, represented with the radical sign \(\sqrt{}\) and responsible for finding a number which, when multiplied by itself, gives the original number under the radical.

• For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).
• For non-perfect squares like \(\sqrt{98}\), we use prime factorization to identify any perfect squares and simplify; \(\sqrt{98}\) is simplified as \(7\sqrt{2}\) since 7 squared is 49.

Radicals maintain their place in expressions because they often cannot be further simplified. That's where simplifying using square roots becomes crucial—like in our exercise—by pulling out perfect square factors, thus reducing the complexity of the expression. This is how weakening the radicand using square roots can turn an intimidating expression like the original \(\sqrt{98} - \sqrt{50} - \sqrt{72}\) into a simple calculation with like terms resulting in \(-4\sqrt{2}\).