Simplify $\sqrt{80}+\sqrt{2 \frac{2}{9}}$
Give your answer in the form $\frac{d \sqrt{e}}{3}$ where $d$ and $e$ are integers.
Answer (1)
Simplify 80 to 4 5 .
Convert 2 9 2 to 9 20 and simplify 2 9 2 to 3 2 5 .
Add the simplified radicals: 4 5 + 3 2 5 = 3 14 5 .
Express the result in the form 3 d e , where d = 14 and e = 5 , so the final answer is 3 14 5 .
Explanation
Initial Analysis First, we need to simplify the expression 80 + 2 9 2 .
Simplifying 80 Let's simplify 80 first. We can rewrite 80 as 16 × 5 , so 80 = 16 × 5 = 16 × 5 = 4 5 .
Simplifying 2 9 2 Now, let's simplify 2 9 2 . First, convert the mixed number to an improper fraction: 2 9 2 = 9 2 × 9 + 2 = 9 18 + 2 = 9 20 . So, 2 9 2 = 9 20 = 9 20 = 3 4 × 5 = 3 2 5 .
Adding the Simplified Radicals Now we can add the simplified radicals: 4 5 + 3 2 5 . To add these, we need a common denominator, which is 3. So we rewrite 4 5 as 3 12 5 . Then we have 3 12 5 + 3 2 5 = 3 12 5 + 2 5 = 3 14 5 .
Final Answer The expression is now in the form 3 d e , where d = 14 and e = 5 .
Examples
Radicals are often used in engineering to calculate lengths and distances. For example, when designing a bridge, engineers use radicals to determine the length of support cables based on the height and span of the bridge. Simplifying radical expressions allows for more precise calculations, ensuring the structural integrity of the bridge.
