Hi Sorry I'm reposting this question, I think the answer that was provided was not correct because of the ambiguity of the original question but if I could have someone workout the question as written it would be much appreciated:
See below discussion.
4x^3y^-2 / 2x^-1y^4
is the answer:
2x^4y^-6 ?
Math - Gray, Saturday, November 20, 2010 at 4:51pm
((4x^3)(y^-2))/((2x^-1)(y^4) The way you typed your equation leaves some ambiguity; I'm assuming this is it. First move the variables with negative exponents to opposite sides of the fraction bar, and make those exponents positive. Then add the exponents and multiply the constants for each side. The answer is (8x^4)/(y^6).
Math - Tia, Saturday, November 20, 2010 at 5:45pm
The way I wrote it is exactly how the book has it. It's all written together without brackets. Very ambiguous I agree but I'm now not sure if the answer you provided is correct.
There is an "art" into transcribing algebraic equations to one single line, as you have done.
I assume the original question in the book was NOT written on one line, but in fractional form, which explains why there are not parentheses.
The line in the middle of the fraction implies that the quantities above and below are to be calculated before the division.
When transcribing to a single line, numerators and denominators that exceed one single term should be enclosed in parentheses, for example:
4 x³ y-2
---------------------
2 x-1 y⁵
should be transcribed as:
(4 x³ y-2) / (2 x-1 y⁵)
to avoid ambiguity.
Most of the time, an expression like:
4x^3y^-2 / 2x^-1y^4
is assumed to be a fraction, but there is always a doubt, as in the present case.
If the assumption is correct, Gray's answer is correct.
The only way to confirm the answer is to confirm the question: was it written as a fraction with an expression above and below a line?
To solve the original question, let's break it down step by step:
The original expression is:
(4x^3y^-2) / (2x^-1y^4)
Step 1: Start by simplifying the individual terms in the numerator and denominator.
In the numerator, we have 4x^3y^-2.
To simplify this term, consider the exponent rules:
- Multiplying two terms with the same base, we add the exponents. So, x^3 * x^-1 = x^(3+(-1)) = x^2.
- For y^(-2), we can bring it down to the denominator and make it positive: y^(-2) = 1/y^2.
So the numerator simplifies to 4x^2 / y^2.
In the denominator, we have 2x^-1y^4.
Again, using the exponent rules:
- x^-1 can be written as 1/x^1 = 1/x.
- No simplification is required for y^4.
So the denominator remains the same: 2/x * y^4.
Now, the expression becomes: (4x^2 / y^2) / (2/x * y^4).
Step 2: Next, divide one fraction by the other. Division is the same as multiplying by the reciprocal.
Dividing (4x^2 / y^2) by (2/x * y^4) is equivalent to multiplying by the reciprocal (y^4 / (2/x * y^4)).
The expression becomes: (4x^2 / y^2) * (y^4 / (2/x * y^4)).
Step 3: Simplify further.
In the numerator, y^4 cancels out with y^2 in the denominator, leaving y^2.
In the denominator, we simplify (2/x * y^4) as (2y^4 / x).
The expression becomes: (4x^2 * y^2) / (2y^4 / x).
Step 4: Finally, multiply the numerators and denominators.
Multiplying 4x^2 and x gives 4x^3.
Multiplying y^2 and 2y^4 gives 2y^6.
The expression simplifies to: (4x^3) / (2y^6), which matches your answer: 2x^4y^-6.
Therefore, the answer provided by Gray in the original discussion is correct.