Problem 45 - Algebra and Trigonometry

Question

\(39-48=\) Simplify the expression. $$ \sqrt{245}-\sqrt{125} $$

Step-by-Step Solution

Answer

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

Step 1: Factor each expression inside the square roots

To simplify the expression, break down the numbers inside the square roots into their prime factors. For 245, the prime factors are 5 and 49 (since 49 is a perfect square, which is 7 squared). For 125, the prime factors are 5 and 25 (since 25 is a perfect square, which is 5 squared).

Step 2: Use perfect squares to simplify square roots

For \( \sqrt{245} \), replace with \( \sqrt{5 \times 49} \), which simplifies to \( 7\sqrt{5} \) because 49 is a perfect square. For \( \sqrt{125} \), replace with \( \sqrt{5 \times 25} \), which simplifies to \( 5\sqrt{5} \) because 25 is a perfect square.

Step 3: Perform the subtraction

Now subtract \( 5\sqrt{5} \) from \( 7\sqrt{5} \). Both terms have the common factor \( \sqrt{5} \), so you can subtract the numeric coefficients: \( 7 - 5 = 2 \). This results in \( 2\sqrt{5} \).

Key Concepts

Prime Factorization

Prime factorization is a technique used to simplify numbers by breaking them down into their basic building blocks, which are prime numbers. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. When you express a number as the product of its prime factors, you make it easier to work with, especially when dealing with square roots. Prime factorization is helpful when simplifying square roots, since it allows us to find perfect squares within the factors, helping to simplify the expression later. For instance, to simplify \( \sqrt{245} \), we identify 245's prime factors as 5 and 49. Since 49 is a perfect square (\( 49 = 7^2 \)), it can be easily factored out.

• Identify the smallest prime number that can divide the given number.
• Divide the number by this prime number and continue the process with the quotient.
• Stop when the quotient itself becomes a prime number.

Using prime factorization, complex expressions inside square roots can be simplified in a structured way.

Perfect Squares

Perfect squares are numbers that are squares of integers. In other words, a perfect square is the product obtained when a whole number is multiplied by itself. Identifying perfect squares is key when simplifying square roots because they tell us which numbers can "come out" of the square root, making the expression simpler.For example, 49 is a perfect square since it equals \( 7^2 \). Therefore, in the square root \( \sqrt{49} \), the term can be simplified to 7, simply because it's a perfect square. The presence of perfect squares within the prime factorization of a number allows the square root itself to be simplified easily.

• Recognize integers whose squares give the perfect square number.
• Pull out the integer from under the square root corresponding to the perfect square.

By identifying and using perfect squares, we can streamline the process of simplifying square roots in mathematical expressions.

Square Root Properties

Square root properties are rules that help simplify expressions under the radical sign, making calculations straightforward and more manageable. One crucial property is that the square root of a product is the product of the square roots of each factor; mathematically shown as \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).This property is particularly useful when simplifying expressions by separating factors within a square root. It allows us to factor out perfect squares, thereby simplifying the expression. For instance, when simplifying \( \sqrt{245} \), recognizing it as \( \sqrt{5 \times 49} \) enables us to use the square root properties to separate and simplify to \( 7\sqrt{5} \). Similarly, \( \sqrt{125} = \sqrt{5 \times 25} \) becomes \( 5\sqrt{5} \).

• Use properties to split square roots into manageable parts.
• Simplify further by reducing perfect squares outside the square root.

Understanding and applying these square root properties make it possible to simplify square root expressions effectively, as illustrated in the given exercise.