Can someone please help me with this problem. I need to simplify this complex rational expression and find the restrictions for X
4+1/x divided by 16-1/xsquared
Thank you
x better not be zero
do you have
(4 + 1/x) / (16 - 1/x^2 ) ????????
if so multiply top and bottom by x^2
(4 x^2 - x) / (16 x^2 - 1)
or
x (4x-1) / [ (4x-1)(4x+1) ]
which is x/(4x+1)
then x also better not be -1/4
also fix the typo moving from
4 + 1/x
to
4x^2 - x
and then proceed to the correct solution...
To simplify the given complex rational expression (4 + 1/x)/(16 - 1/x^2), we can follow these steps:
Step 1: Find the least common denominator (LCD).
In this case, the LCD is x^2, as it is the least common multiple of x and x^2.
Step 2: Rewrite each fraction with the LCD as the denominator.
(4 + 1/x)/(16 - 1/x^2) = [(4x + 1)/x]/[(16x^2 - 1)/(x^2)]
Step 3: Change the division into multiplication by taking the reciprocal of the second fraction.
[(4x + 1)/x] * [(x^2)/(16x^2 - 1)]
Step 4: Cross-cancel common factors.
In this case, we observe that there is x in the numerator and denominator, so we can cross-cancel:
[(4 + 1/x)] * [(x)/(16x^2 - 1)]
Step 5: Multiply the remaining terms.
(4 + 1/x)(x)/(16x^2 - 1) = (4x + 1)/(16x^2 - 1)
Therefore, the simplified expression is (4x + 1)/(16x^2 - 1).
Now let's find the restrictions for x.
To determine the restrictions, we need to consider the denominator of the simplified expression, which is (16x^2 - 1).
The denominator cannot equal zero, as division by zero is undefined. So, we set the denominator equal to zero and solve for x:
16x^2 - 1 = 0
To solve this quadratic equation, we can first add 1 to both sides:
16x^2 = 1
Then divide both sides by 16:
x^2 = 1/16
To determine the value(s) of x, we take the square root of both sides:
x = ±√(1/16)
Simplifying the square root:
x = ±(1/4)
Therefore, the restrictions for x are x ≠ -1/4 and x ≠ 1/4.