Problem 29 - Algebra and Trigonometry

Question

Simplify the expression. \(\sqrt{245}-\sqrt{125}\)

Step-by-Step Solution

Answer

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

Step 1: Factorize the Radicands

First, factor each radicand into its prime factors. Start with 245: \[ 245 = 5 \times 49 = 5 \times 7^2 \] Similarly, factor 125: \[ 125 = 5 \times 25 = 5 \times 5^2 \]

Step 2: Simplify the Square Roots

Using the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \), simplify each square root. For \( \sqrt{245} \):\[ \sqrt{245} = \sqrt{5 \times 7^2} = \sqrt{5} \times \sqrt{7^2} = 7\sqrt{5} \] For \( \sqrt{125} \):\[ \sqrt{125} = \sqrt{5 \times 5^2} = \sqrt{5} \times \sqrt{5^2} = 5\sqrt{5} \]

Step 3: Combine Like Terms

Subtract the terms with the same radical part by identifying like terms. So, \( 7\sqrt{5} - 5\sqrt{5} \) can be simplified to: \[ (7 - 5)\sqrt{5} = 2\sqrt{5} \]

Key Concepts

Prime Factorization

Prime factorization is an essential technique when dealing with simplifying radical expressions. It involves expressing a number as a product of its prime factors, which helps simplify calculations. To prime factorize a number, continuously divide the number by the smallest prime number until you reach 1.

• For 245, we start by dividing by 5, obtaining 49. Since 49 is not divisible by 5, we move to the next smallest prime number, 7. We divide 49 by 7 to get 7 again. So, the prime factorization of 245 is: \[245 = 5 \times 7^2\]
• For 125, we consistently divide by 5, which leads us to: \[125 = 5 \times 5^2\]

Once each radicand is expressed in terms of its prime factors, it becomes straightforward to simplify the square roots.

Square Roots

The square root of a number is another value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 multiplied by 3 equals 9. When simplifying expressions involving square roots, leveraging the property that \[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \] is helpful.This means you can "break down" a square root into the square roots of its prime factors.

• For \(\sqrt{245}\), since the prime factorization is \(5 \times 7^2\), you can separate this into \(\sqrt{5}\) and \(\sqrt{7^2}\). Because the square root of 7 squared is 7, \(\sqrt{245}\) simplifies to 7\(\sqrt{5}\).
• Similarly, for \(\sqrt{125}\), the prime factors give \(\sqrt{5 \times 5^2}\). Again, the square root of 5 squared is 5, simplifying \(\sqrt{125}\) to 5\(\sqrt{5}\).

Like Terms

In math, like terms refer to terms whose algebraic components (variables and exponents) are identical. When combining like terms, you combine the coefficients while keeping the common part intact. In the context of the exercise, the terms 7\(\sqrt{5}\) and 5\(\sqrt{5}\) are considered like terms because they share the same radical part, \(\sqrt{5}\).

• To combine these terms, subtract the coefficients of \(\sqrt{5}\) as they share the same radical. Thus, subtract 5 from 7 to get 2.
• The expression \(7\sqrt{5} - 5\sqrt{5}\) becomes \(2\sqrt{5}\).

Combining like terms is crucial in simplifying algebraic expressions efficiently, making complex calculations more manageable.