Simplifying square roots involves finding the largest perfect square that divides evenly into the number under the square root symbol. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25). We then rewrite the square root as the product of the square root of the perfect square and the square root of the remaining factor. Finally, we take the square root of the perfect square and write it as a coefficient of the remaining square root. This process makes it easier to work with and understand irrational numbers. For example, to simplify $\sqrt{12}$, find the largest perfect square factor (4). Thus, $\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}$.
Why do we simplify? It allows us to combine like terms when adding or subtracting radicals. For instance, $2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}$. Without simplification, this combination would be difficult.
Match the following terms with their definitions:
| Term | Definition |
|---|---|
| 1. Perfect Square | A. A number that cannot be written as a fraction. |
| 2. Radicand | B. A number that when multiplied by itself equals a given number. |
| 3. Irrational Number | C. The number or expression inside the radical symbol. |
| 4. Square Root | D. A number that is the result of squaring an integer. |
| 5. Coefficient | E. A number multiplied by a variable or radical. |
(Answers: 1-D, 2-C, 3-A, 4-B, 5-E)
Complete the following paragraph using the words provided below:
To simplify a square root, we look for the largest _________ _________ factor of the _________. For example, to simplify $\sqrt{50}$, we find that 25 is the largest perfect square that divides 50. Therefore, $\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$. The simplified form is $5\sqrt{2}$, where 5 is the _________ and 2 is the remaining factor inside the square root.
(Words: perfect square, radicand, coefficient)
Explain why simplifying square roots is a useful skill in algebra. Give an example where it makes a problem easier to solve.
Please log in to post your answer.
Log InEarn 2 Points for answering. If your answer is selected as the best, you'll get +20 Points! 🚀