A factor tree branches down into 2 branches. The first number 1,225 branches down on the left to 5 and on the right to 245. 245 branches down to 5 on the left and 49 on the right. 49 branches down to 7 on the left and 7 on the right. A factor tree is shown for 1,225. What is the simplest form of √1,225?
The Correct Answer and Explanation is:
To simplify 1,225\sqrt{1,225}1,225, we can use the factor tree that you’ve described. Let’s break it down step by step:
Factor Tree Breakdown:
- 1,225 branches down into 5 on the left and 245 on the right. 1,225=5×2451,225 = 5 \times 2451,225=5×245
- 245 branches down into 5 on the left and 49 on the right. 245=5×49245 = 5 \times 49245=5×49
- 49 branches down into 7 on the left and 7 on the right. 49=7×749 = 7 \times 749=7×7
Full Prime Factorization:
From the tree, we can express 1,2251,2251,225 as:1,225=5×5×5×7×71,225 = 5 \times 5 \times 5 \times 7 \times 71,225=5×5×5×7×7
This simplifies to:1,225=53×721,225 = 5^3 \times 7^21,225=53×72
Simplifying the Square Root:
Now, to simplify 1,225\sqrt{1,225}1,225, we use the property of square roots that a×b=a×b\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}a×b=a×b. Applying this to the factorization:1,225=53×72\sqrt{1,225} = \sqrt{5^3 \times 7^2}1,225=53×72
We can split the square root into two parts:1,225=52×5×72=52×72×5\sqrt{1,225} = \sqrt{5^2 \times 5 \times 7^2} = \sqrt{5^2} \times \sqrt{7^2} \times \sqrt{5}1,225=52×5×72=52×72×5
Since 52=5\sqrt{5^2} = 552=5 and 72=7\sqrt{7^2} = 772=7, this simplifies to:1,225=5×7×5=355\sqrt{1,225} = 5 \times 7 \times \sqrt{5} = 35 \sqrt{5}1,225=5×7×5=355
Final Answer:
The simplest form of 1,225\sqrt{1,225}1,225 is:355\boxed{35\sqrt{5}}355
Explanation:
The square root of 1,2251,2251,225 cannot be simplified further because 5\sqrt{5}5 is an irrational number. Thus, the simplest exact form is 35535\sqrt{5}355, which provides a clean representation of the square root.
