How can you rewrite 9 1/3

Question

How can you rewrite  9 1/3
  using a root

GPT-4o mini · Answer

To rewrite $ 9 \frac{1}{3} $ using a root, first convert the mixed number into an improper fraction. A mixed number consists of a whole number and a fraction. The whole number part here is 9 and the fractional part is $ \frac{1}{3} $. 1. Convert $ 9 \frac{1}{3} $ to an improper fraction: $$ 9 \frac{1}{3} = \frac{9 \times 3 + 1}{3} = \frac{27 + 1}{3} = \frac{28}{3} $$ Now that we have $ \frac{28}{3} $, we can look for ways to express this using a root. One approach is to express it in terms of a square root or cube root. For example, you can express $ 28 $ using a square root as follows: $$ \frac{28}{3} = \frac{4 \cdot 7}{3} = \frac{4 \sqrt{7^2}}{3} = \frac{4 \sqrt{49}}{3} $$ However, this doesn't present a neat root expression. Alternatively, you could express it in terms of roots in a simpler way such as making use of the square root of $ 28 $: $$ \frac{28}{3} = \frac{28}{3} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{28 \sqrt{3}}{3 \sqrt{3}} = \frac{28 \sqrt{3}}{9} $$ This maintains the value of $ 9 \frac{1}{3} $. Thus, while directly rewriting $ 9 \frac{1}{3} $ using a root is mathematically tricky, you can express its equivalent in forms involving roots.

bluesanta · Answer

whats the answer using root

GPT-4o mini · Answer

To express $ 9 \frac{1}{3} $ using a square root, you can represent the mixed number as an improper fraction and then use a root notation. Starting from the mixed number: $$ 9 \frac{1}{3} = \frac{28}{3} $$ To rewrite this using a square root, consider expressing $ 28 $ in a way that involves a root: $$ 28 = 4 \cdot 7 = \sqrt{16} \cdot 7 = \sqrt{16 \times 7} $$ So we could express: $$ 9 \frac{1}{3} = \frac{28}{3} = \frac{\sqrt{16 \cdot 49}}{3} $$ However, to write $ \frac{28}{3} $ as a single fraction with a root might not simplify neatly. To keep it straightforward, one practical way is: $$ 9 \frac{1}{3} = \frac{28}{3} \approx 9.33 $$ A more natural approach would be an approximate form, such as: $$ 9 \frac{1}{3} \approx \sqrt{87.11} $$ This value will not be exact but indicates that $$ 9 \frac{1}{3} \approx \sqrt{87.11} $$ But to strictly answer your question about using a root, still, sometimes the best method is simply keeping the improper fraction: Thus, $ 9 \frac{1}{3} $ can't be nicely expressed solely in terms of basic roots without losing exactness. If you want a specific expression that directly shows a root while maintaining similarity, a viable one isn't straightforward based on just pure root expressions, as the decimal form communicates the value better. Again, for practical purposes of using a root for $ 9 \frac{1}{3} $ in simplest terms: $$ \frac{28}{3} = 9.33 \text{ (exact) or }\sqrt{87.11} \text{ (approx)} $$ This indicates roots don't directly translate neatly into mixed numbers like this without approximations. If clarification is needed on the numeric root form or different approaches, please ask!

bluesanta · Answer

What is the simplified form of 2713

GPT-4o mini · Answer

The number $ 2713 $ is already in its simplest form as it is a whole number. However, if you are looking to simplify it in terms of prime factorization, we can break it down into its prime factors. To find the prime factors of $ 2713 $: 1. **Check for divisibility by small prime numbers** (like 2, 3, 5, 7, etc.): - $ 2713 $ is odd, so it's not divisible by $ 2 $. - To check if it is divisible by $ 3 $, add the digits $ 2 + 7 + 1 + 3 = 13 $, which is not divisible by $ 3 $. - It does not end in $ 0 $ or $ 5 $, so it is not divisible by $ 5 $. - For $ 7 $: $ 2713 ÷ 7 ≈ 387.57 $ (not an integer). - For $ 11 $: $ 2713 ÷ 11 ≈ 246.64 $ (not an integer). - Continuing this, you will find that $ 2713 $ is not divisible by primes up to about $ 53 $ (the approximate square root of $ 2713 $). After testing for small prime numbers, it turns out that $ 2713 $ is a prime number itself. So, the simplified, or prime factorization form of $ 2713 $ is simply: $$ 2713 \text{ (as it is a prime number)} $$ If you intended something different by "simplified form", please provide more context!