Problem 182 - Elementary Algebra

Question

In the following exercises, simplify. $$ \sqrt{72}-\sqrt{98} $$

Step-by-Step Solution

Answer

-\(\sqrt{2}\).

Step 1: Simplify each square root

First, simplify the square root of each term. \[\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}.\]Next, simplify \(\sqrt{98} = \sqrt{49 \cdot 2} = \sqrt{49} \cdot \sqrt{2} = 7\sqrt{2}\).

Step 2: Substitute simplified terms

Now substitute the simplified forms into the original expression: \(6\sqrt{2} - 7\sqrt{2}\)

Step 3: Combine like terms

Finally, combine like terms to get the simplified expression: \(6\sqrt{2} - 7\sqrt{2} = -1\sqrt{2} = -\sqrt{2}\).

Key Concepts

Square Root Simplification

To simplify square roots, you should decompose the number inside the square root into its factors, ideally including a perfect square.
For example, simplify \(\sqrt{72}\) by finding two numbers that multiply to 72, one of which should be a perfect square. In this case, 72 can be expressed as 36 \times 2. Since 36 is a perfect square, \(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}.\)
Similarly, \(\sqrt{98}\) can be broken into factors 49 \times 2. With 49 being a perfect square, you get \(\sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \cdot \sqrt{2} = 7\sqrt{2}.\)
Breaking numbers into their perfect square factors simplifies square roots significantly.

Combining Like Terms

Combining like terms is a crucial skill in algebra. It involves simplifying an expression by adding or subtracting terms that have identical variables and exponents.
Take the simplified terms from our example: \(6\sqrt{2} - 7\sqrt{2}.\)
Since both terms contain \(\sqrt{2},\) they are like terms and can be combined.
You adjust the coefficients (numbers in front), so: \(6\sqrt{2} - 7\sqrt{2} = (6 - 7)\sqrt{2} = -1\sqrt{2}.\)
Therefore, our simplified expression is \(-\sqrt{2}.\)
Always double-check to ensure the terms you're combining have the same radical parts to avoid mistakes.

Radical Expressions

A radical expression includes a root, such as a square root, cube root, etc.
In our example, \(\sqrt{72} - \sqrt{98},\) we deal with square roots.
Simplify each term first, then combine them. Simplifying these expressions helps in performing operations like addition or subtraction on them.
For example, converting \(\sqrt{72}\) to \(6\sqrt{2}\) and \(\sqrt{98}\) to \7\sqrt{2}\ makes it easier to see the next step: \(6\sqrt{2} - 7\sqrt{2}.\)
Radicals can often be intimidating, but breaking them down step by step makes them much more manageable.
Regular practice with radical expressions can help solidify these skills, making them easier to work with over time.