Simplify the following number.
∛729
I think it is ∛9, but I do not know. Please help, and check my answer. I will appreciate it if you guys can answer this as soon as possible. It is part of a RATIONAL EXPONENTS question.
The original equation was 9^(1/3)∙〖81〗^(1/3). Once I turned both into square roots, I multiplied them, and I got ∛729, which I think simplifies to ∛9. Also, please again answer and check my answer, and answer as soon as possible. I will appreciate it if you do.
since 729 = 9^3, you are correct.
Your method was also correct.
81 = 9^2, so
81^(1/3) = (9^2)^(1/3) = 9^(2/3)
and
9^(1/3) * 9^(2/3) = 9^(3/3) = 9
fractional exponents are just like any others. Add them when multiplying, and multiply them when raising powers to powers.
oops, the answer is 9, not ∛9
9 is in fact ∛729
Thank you, I just wasn't sure if it needed to be simplified more, or I needed to do it again by doing another way. So, everything leads to 9, I'll remember that.
To simplify the expression ∛729, let's start by finding the prime factorization of 729.
Prime factorization is the process of breaking down a number into its prime factors. Since 729 is a perfect cube, we can factor it by repeatedly dividing it by prime numbers until we cannot divide it any further.
The prime factorization of 729 is:
729 ÷ 3 = 243
243 ÷ 3 = 81
81 ÷ 3 = 27
27 ÷ 3 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
So, the prime factorization of 729 is 3^6.
Now, since we have ∛729, we can rewrite it as (∛3^6).
When we take the cube root of a number raised to an exponent, we can simplify it by dividing the exponent by 3. In this case, since the exponent is 6, we divide it by 3:
6 ÷ 3 = 2
Therefore, ∛729 simplifies to 3^2, which is equal to 9.
So, your initial intuition was correct! The simplified form of ∛729 is 9.