1.1.21 simplify rational expression

In summary: Fractions such as 1/2, 1/3, 1/4, etc. use the same numerals as do negative exponents. That does not make them the same thing.In summary, the given expression $\left(\dfrac{a^2b^3-2a^{-3}b^3}{2a}\right)^2$ can be simplified by cancelling out the common factor of "a" before squaring. This results in the expression $\frac{\left(ab^3-2a^{-4}b^3\right)^2}{4}$ which can be further simplified by factoring out the common factor of $b^3$ and then expanding the squared term. The
  • #1
karush
Gold Member
MHB
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simplify
$\left(\dfrac{a^2b^3-2a^{-3}b^3}{2a}\right)^2=$
OK this could get confusing quickly
but I don't think we want to square it first
sscwt.png
 
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  • #2
Yes, it can be simplified by cancelling "a" before squaring.
(The "2" in the denominator is already simple.)
$\frac{\left(ab^3-2a^{-4}b^3\right)^2}{4}$

Of course we can also factor out that $b^3$ that is in each term. Squaring it gives $b^6$.

$\frac{b^6(a- 2a^{-4})^2}{4}$

Now it is relatively easy to square $a- 2a^{-4}$.
$(a- 2a^{-4})^2= a^2- 2(a)(2a^{-4})+ (2a^{-4})^2$
$= a^2- 4a^{-3}+ 4a^{-8}$

$\frac{b^6(a^2-4a^{-3}+4a^{-8}}{4}$.
 
  • #3
ok we still have negative exponents in the numerator??
 
  • #4
Is that a problem? Different people have different ideas of what "simple" is. If you don't want negative exponents, remember what a negative exponent means:
$\frac{b^2- 4a^{-3}+ 4a^{-8}}{4}= \frac{b^2- 4\frac{1}{a^{3}}+ 4\frac{1}{a^{8}}}{4}$

Now, to eliminate the "fractions in fractions", multiply both numerator and denominator by $a^8$
$\frac{a^8b^3- 4a^5+ 4}{4a^8}$
 
  • #5
well the few examples I saw they brought the terms with negative exponents in the numerator to the denumerator
 
  • #6
That is exactly what I did!
 
  • #7
Country Boy said:
That is exactly what I did!

yes but you created fractions over over a fraction $ \dfrac{a^{-3}}{4}=\dfrac{1}{4a^3}$

anyway,, just being cranky
 
  • #8
No, [tex]a^{-3}[/tex] is not a fraction!
 
  • #9
why not the exponent is positive
 
  • #10
Country Boy said:
No, [tex]a^{-3}[/tex] is not a fraction!

It is when it's written as $\displaystyle \begin{align*} \frac{1}{a^3} \end{align*}$...
 
  • #11
Yes, $\frac{1}{x^3}$ is a fraction. $x^{-3}$ is NOT.
It is a matter of the difference in numerals, not numbers.
 
  • #12
numerals?
 
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  • #13
Country Boy said:
Yes, $\frac{1}{x^3}$ is a fraction. $x^{-3}$ is NOT.
It is a matter of the difference in numerals, not numbers.

Do you realize that you are saying that two amounts that are equivalent are simultaneously a fraction and not a fraction?
 
  • #14
they just said move the terms with negative exponents in the numerator to the denominator
the bell rang and I left
no fancy stuff
 
  • #15
Prove It said:
Do you realize that you are saying that two amounts that are equivalent are simultaneously a fraction and not a fraction?
Yes, I am. They are two different numerals that repreent the same number. The term "fraction" has to do with the way a number is represented, it is not a property of the number itself.
 

Related to 1.1.21 simplify rational expression

What is a rational expression?

A rational expression is an algebraic expression that contains one or more fractions with variables in the numerator and/or denominator.

How do you simplify a rational expression?

To simplify a rational expression, you need to factor both the numerator and denominator, then cancel out any common factors. This will result in a simplified expression with no common factors remaining.

Can a rational expression have a variable in the denominator?

Yes, a rational expression can have a variable in the denominator. However, when simplifying, you must be careful to avoid dividing by zero, so any values that would make the denominator equal to zero must be excluded from the domain of the expression.

What is the difference between simplifying and solving a rational expression?

Simplifying a rational expression involves reducing it to its simplest form by canceling out common factors. Solving a rational expression, on the other hand, involves finding the value(s) of the variable(s) that make the expression equal to a given value.

What are some common mistakes to avoid when simplifying a rational expression?

Some common mistakes to avoid when simplifying a rational expression include forgetting to factor completely, canceling out non-common factors, and dividing by zero. It is also important to check your final answer to ensure it is in its simplest form and that all excluded values have been accounted for.

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