Problem 59 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(-4\sqrt{2}\).

Step 1: Factor Inside the Radicals

First, factor the numbers inside each square root to find perfect squares. \(98 = 49 \times 2\), so \(\sqrt{98} = \sqrt{49 \times 2} \).\(50 = 25 \times 2\), so \(\sqrt{50} = \sqrt{25 \times 2} \).\(72 = 36 \times 2\), so \(\sqrt{72} = \sqrt{36 \times 2} \).

Step 2: Simplify Each Radical

Next, simplify each square root by taking the square root of perfect squares:\(\sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2}\).\(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\).\(\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}\).

Step 3: Combine Like Radicals

Now, combine the radicals by combining their coefficients:\(7\sqrt{2} - 5\sqrt{2} - 6\sqrt{2} = (7 - 5 - 6)\sqrt{2}\).

Step 4: Calculate the Final Coefficient

Subtract the coefficients:\(7 - 5 - 6 = -4\).

Step 5: Express the Final Simplified Expression

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

Key Concepts

Combining Like Terms

Combining like terms is a core concept when working with mathematical expressions. In the context of radicals, especially when simplifying expressions with square roots, it is crucial to identify like terms and combine them effectively.

In our example,

• we have three square root terms:
  • \(\sqrt{98} = 7\sqrt{2}\)
  • \(\sqrt{50} = 5\sqrt{2}\)
  • \(\sqrt{72} = 6\sqrt{2}\).

All these radicals contain \(\sqrt{2}\) and can hence be combined. The general idea is to treat the coefficients in front of the like radicals as numbers we can add or subtract just like any other. In this case, combining the terms means taking the expression,

• \(7\sqrt{2} - 5\sqrt{2} - 6\sqrt{2}\),

and performing simple arithmetic with the coefficients:

• \(7 - 5 - 6 = -4\).

Thus, the simplified expression becomes \(-4\sqrt{2}\), elegantly merging terms via their common square root component.

Square Roots

Square roots are fundamental in algebra, representing one of the most common operations you'll encounter. Simplifying expressions with square roots involves identifying components that can be rewritten more succinctly.

With square roots such as \(\sqrt{98}\), we explored simplifying it by first factoring the number inside:

• \(98 = 49 \times 2\).

From the factorization, we derived

• \(\sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \times \sqrt{2} = 7\sqrt{2}\).

Similarly, for \(\sqrt{50}\) and \(\sqrt{72}\), we identified perfect square factors:

• \(25\) in \(50\), resulting in \(5\sqrt{2}\),
• and \(36\) in \(72\), giving \(6\sqrt{2}\).

This process of breaking down numbers under the square root into their prime or perfect square factors allows us to see potential subtractions or additions involving radicals more clearly. The end goal with square roots is to identify and separate these to better combine like radicals later.

Factoring Inside the Radical

Factoring is a powerful tool for simplifying radicals. It involves re-writing numbers under the root as the product of their factors, with an eye towards finding perfect squares.

In working with \(\sqrt{98}\), \(\sqrt{50}\), and \(\sqrt{72}\), breaking these down into easily managed components was essential:

• Finding factors like \(49\), \(25\), and \(36\), are key because they are perfect squares.

Perfect squares simplify neatly and let you bring values out from under the radical. For instance,

• \(\sqrt{98} = \sqrt{49 \times 2}\) transforms to \(7\sqrt{2}\) because \(\sqrt{49} = 7\).

The essence of factoring inside the radical lies in recognizing and utilizing these perfect squares. These steps reduce larger radicals step-by-step into simpler, more manageable parts that invite direct arithmetic operations, like adding or subtracting like terms.