Question Video: Simplifying Algebraic Expressions Using Laws of Exponents with Negative Exponents Mathematics

Video Transcript

Simplify 𝑥 divided by 𝑥 squared, given that 𝑥 does not equal zero.

In this question, we are asked to simplify a given expression using the fact that the variable 𝑥 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 𝑥 over 𝑥 squared. We can then see that we have the quotient of two expressions involving 𝑥. In fact, we can note that this is the quotient of two exponential expressions with a base of 𝑥 by recalling that raising any number to the first power leaves it unchanged. So, 𝑥 equals 𝑥 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 𝑏 raised to the power of 𝑚 over 𝑏 raised to the power of 𝑛 is equal to 𝑏 raised to the power of 𝑚 minus 𝑛. In other words, we raise the base to the difference in the exponents. In our expression, our value of 𝑏 is 𝑥, our value of 𝑚 is one, and our value of 𝑛 is two. So, we have 𝑥 raised to the power of one minus two. We can then evaluate the expression in the exponent to obtain 𝑥 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, 𝑥 over 𝑥 squared is equal to 𝑥 over 𝑥 times 𝑥. Now, since 𝑥 is nonzero, we can cancel the shared factor of 𝑥 in the numerator and denominator to obtain one over 𝑥.

We could leave our answer as one over 𝑥. However, we can rewrite this as an exponential expression by recalling the negative exponent rule, which tells us that one over 𝑏 raised to the power of 𝑛 is equal to 𝑏 raised to the power of negative 𝑛. We can rewrite 𝑥 in the denominator as 𝑥 raised to the first power and use the negative exponent rule to once again obtain an answer of 𝑥 raised to the power of negative one.