Video Transcript
Simplify π₯ divided by π₯ squared,
given that π₯ does not equal zero.
In this question, we are asked to
simplify a given expression using the fact that the variable π₯ is nonzero. To do this, we can start by noting
that the given expression is a division of two expressions. Although it is not necessary, we
can start by rewriting this division in the more conventional notation of π₯ over π₯
squared. We can then see that we have the
quotient of two expressions involving π₯. In fact, we can note that this is
the quotient of two exponential expressions with a base of π₯ by recalling that
raising any number to the first power leaves it unchanged. So, π₯ equals π₯ raised to the
first power.
We can then recall that the
quotient rule for exponents allows us to simplify the quotient of two exponential
expressions with the same nonzero base. It tells us that π raised to the
power of π over π raised to the power of π is equal to π raised to the power of
π minus π. In other words, we raise the base
to the difference in the exponents. In our expression, our value of π
is π₯, our value of π is one, and our value of π is two. So, we have π₯ raised to the power
of one minus two. We can then evaluate the expression
in the exponent to obtain π₯ raised to the power of negative one.
It is worth noting that this is not
the only way that we can answer this question. We can recall that squaring a
number means multiplying it by itself, so, π₯ over π₯ squared is equal to π₯ over π₯
times π₯. Now, since π₯ is nonzero, we can
cancel the shared factor of π₯ in the numerator and denominator to obtain one over
π₯.
We could leave our answer as one
over π₯. However, we can rewrite this as an
exponential expression by recalling the negative exponent rule, which tells us that
one over π raised to the power of π is equal to π raised to the power of negative
π. We can rewrite π₯ in the
denominator as π₯ raised to the first power and use the negative exponent rule to
once again obtain an answer of π₯ raised to the power of negative one.
