What is the Understood 1?

What happens when you multiply by 1?

What happens when you multiply or divide something by $1$, or raise something to the first power? It stays the same. This is the premise of the “understood $1$.”

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When you have a variable such as $x$, you should get to a point where you almost immediately remember that “just plain $x$” can be written as $1x$, because $x=1\cdot x=1x$, and similarly that $x$ can be written as $x/1$ or $x^1$.

Cases of the Understood $1$:

$x=1\cdot x$

$x=1x$

$x=\frac{x}{1}$

$x=x^1$

If we combine the kinds of techniques that we used to get these expression, we can also say

$x=\frac{1x^1}{1}$

Why is this useful? There are times when $x$ appears in a fraction, or $x$ is added to some term in which $x$ appears, or $x$ is multiplied by some factor in which $x$ appears. When something like this happens, it’ll be helpful to remember the understood $1$ (and maybe even write out a $1$ instead of leaving it as an understood $1$).

Applying the understood 1 as a coefficient, an exponent, and a denominator

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How to use the idea of the understood 1 to simplify expressions

Example

Simplify the expression.

$(x +2x) \cdot x^3$

Start by simplifying parentheses and remember that $x=1x$.

$(1x+2x)\cdot x^3$

$(3x)\cdot x^3$

Multiply and remember that $3x=3x^1$.

$3x^1\cdot x^3$

$3x^4$

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This time let’s go in the opposite direction, by looking at an example where one or more “unnecessary” $1$’s appear in an expression that we can simplify by replacing the unnecessary $1$’s with understood $1$’s.

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“just plain x” can be written as 1x, or as x/1, or as x^1.

Example

Simplify the expression.

$\frac{1}{1(1x^1)}+1\left(\frac{x}{1}+1\right)$

In this example the understood $1$ has been written out multiple times. Let’s simplify by removing the understood $1$.

$\frac{1}{1x^1}+1\left(\frac{x}{1}+1\right)$

$\frac{1}{x^1}+1\left(\frac{x}{1}+1\right)$

$\frac{1}{x}+1\left(\frac{x}{1}+1\right)$

$\frac{1}{x}+\frac{x}{1}+1$

$\frac{1}{x}+x+1$

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In our next example, we’ll see how to deal with the understood $1$ when we want to add two fractions where the denominator of one fraction is just plain $x$ and the denominator of the other fraction is $x$ raised to some power other than $1$.

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Example

Add the fractions.

$\frac{2}{x}+\frac{5}{x^3}$

In the denominator of the first fraction, use the fact that $x=x^1$.

$\frac{2}{x^1}+\frac{5}{x^3}$

If we multiply the top and bottom of the first fraction by $x^2$, we’ll find that we can use $x^3$ as the common denominator.

$\frac{2}{x^1}\left(\frac{x^2}{x^2}\right)+\frac{5}{x^3}$

$\frac{2(x^2)}{x^1(x^2)}+\frac{5}{x^3}$

$\frac{2x^2}{x^3}+\frac{5}{x^3}$

$\frac{2x^2+5}{x^3}$

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