so i got a question and would like to know if i did this right
For problems 1 through 38, transform the given expression to simple radical form. You can check your answer by evaluating it and the original expression by calculator.
on the previous page this definition was given
Simple Radical Form
An expression is in simple radical form if
1. the radicand of an nth root contains no nth powers as factors,
2. the root index is as low as possilbe, and
3. there are no radicals in the denominator
and this is problem number 4
4. 108^(1/3) + 10(32)^(1/3)+ 500^(1/3)
and for my answer I got
about 31.4289(2)^(1/2)
my instructors instructions were to round to four decimals out from the decimal which I have done...
Is this answer correct?
If not what rule did I break?
If I did do this wrong how do I do it the right way?
I checked by calculator and got the right answer so I think I have the right answer and I don't think I broke any of the rules according to my textbook...
so I would just like to make sure...
also what is the point of having the division symbol when 3*1/4 can be written as 3 * 4^-1 which makes more sense to write out like to me because then all of the fraction rules actually work that I was taught back in like 4th grade but was not provided any proof...
108^(1/3) + 10(32)^(1/3)+ 500^(1/3)
=(27*4)1/3 + 10*(4*8)1/3 + (125*4)1/3
=(3+20+5)(4)1/3
check my thinking.
makes sense to me but is my answer also correct?
I do not think that your answer was obtained without a calculator. To get SRform, you should have to do no calculations with roots. See my work. If I were your teacher, and you gave me that answer, I would mark it wrong, and let you prove you got it without a calculator.
huh interesting well thanks
my only last question is what's is the point of us SRform?
I undrestand that in some rare cases you might not have a calculator but in most cases you will... but anyways using SRform is to simplify when you don't have a calculator correct?
If this is so what good does it do you if get (3+20+5)(4)1/3 as an answer because you wouldn't be albe to get an answer that you could do anything with correct??? Like if you were trying to find the length of something and had to cut a piece of wood or something you wouldn't know what (3+20+5)(4)1/3 is and wouldn't be able to do anything unless you could evaluate with a calculator correct???
Also what do you recomend one to do if you have a very large number and have to use SRform such as 5.99 E 24 kg the mass of the earth correct? and have to use SR form of something like that???? What would you do without a calculator?
also does it make more sense to write fractions out with inverse exponents instead because for some reason to me it makes a lot more sense????
M+4
M-6
To determine if your answer is correct for problem 4, let's go through the process of transforming the expression to simple radical form.
Starting with the expression: 108^(1/3) + 10(32)^(1/3) + 500^(1/3)
1. First, let's simplify each radical individually:
- 108^(1/3) is the cube root of 108. We can simplify this by finding the perfect cubes that divide evenly into 108. The largest perfect cube that divides evenly into 108 is 27 (which is 3^3), so we can rewrite 108 as 27 * 4. Taking the cube root of 27 gives us 3, so we can rewrite 108^(1/3) as 3 * 4^(1/3).
- Similarly, 32 can be written as 16 * 2, and taking the cube root of 16 gives us 2, so (32)^(1/3) simplifies to 2 * 2^(1/3).
- Lastly, 500 can be written as 125 * 4, and taking the cube root of 125 gives us 5, so 500^(1/3) simplifies to 5 * 4^(1/3).
2. Now, let's write the expression in its simplified form:
3 * 4^(1/3) + 10 * (2 * 2^(1/3)) + 5 * 4^(1/3)
3. Since we have a common term, 4^(1/3), in both the first and last terms, we can combine them:
(3 + 5) * 4^(1/3) + 10 * (2 * 2^(1/3))
4. Simplify further:
8 * 4^(1/3) + 10 * 2 * 2^(1/3)
5. Now, we can simplify 4^(1/3) to its radical form. Remember the rules for the simple radical form:
- The radicand should contain no nth powers as factors (which is satisfied).
- The root index should be as low as possible (which is satisfied).
- There should be no radicals in the denominator (which is satisfied).
Since 4 is a perfect square, we can represent it as 2^2. Therefore, 4^(1/3) can be written as (2^2)^(1/3), which simplifies to 2^(2/3).
6. Plug this simplification back into the expression:
8 * 2^(2/3) + 10 * 2 * 2^(1/3)
7. Finally, we can simplify further by combining the exponents:
8 * 2^(2/3) + 10 * 2^(4/3)
Now, evaluating this expression further would require a calculator or approximation methods. However, your answer of approximately 31.4289(2)^(1/2) does not seem to match this simplification process.
Regarding your question about the division symbol, it is true that 3 * 1/4 can be written as 3 * 4^(-1), where the negative exponent indicates the reciprocal. The division symbol is simply another way to represent the division operation. Both expressions are equivalent and follow the same rules of fraction manipulation.
In summary, to correctly transform expressions into simple radical form, follow the rules stated in your textbook and apply simplification methods step by step. Double-check your calculations to ensure accuracy before submitting your final answer.