# Does a/(bc) = (a/b) * (a/c)?

This is part of a series on common misconceptions.

True or False?\(\frac{a}{bc}\) is equal to \(\frac{a}{b}\) x \(\frac{a}{c}.\)

**Why some people say it's true:** You can multiply fractions like this to get the denominator.

**Why some people say it's false:** The numerator is \(a,\) not \(a^2.\)

The statement is \( \color{red}{\textbf{false}}\).

Proof:When multiplying the second fraction, we get \(\frac{a^2}{bc}.\) However, \(\frac{a}{bc}\)=\(\frac{1}{bc}\) x a or \(\frac{1}{b}\) x \(\frac{a}{c}\) or \(\frac{\hspace{2mm} \frac a2\hspace{2mm} }{b}\) x \(\frac{2}{c},\) etc. \(_\square\)

Rebuttal: But if \(a=1,\) it works.

Reply: But it doesn’t work for all values of \(a.\)

Rebuttal: What if \(a\) stands for a different number each time?

Reply: That would break the rules of algebra.

Rebuttal: What if \(a\) stands for the square root of anything? \(i\) can be \(+i\) or \(- i\) and then \(a\) can be different values.

Reply: \(i\) is just \(+i\) and \(a\) can stand for one value only.

Rebuttal: So it never works?

Reply: It works if \(a=1\) or \(a=0.\)

Rebuttal: How can we judge the value if \(b=0\) or \(c=0?\)

Reply: No value could be given to the final fraction, so we can’t do anything, although if \(a=2, b=6,\) and \(c=4,\) then the statement doesn’t work anyway.

**See Also**

**Cite as:**Does a/(bc) = (a/b) * (a/c)?.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/is-split-division/