Question Video: Evaluating an Expression Involving a Negative Integer Exponent Mathematics • Second Year of Preparatory School

Video Transcript

If 𝑥 is equal to root four over root two, which of the following is equal to 𝑥 to the power of negative one? (A) One, (B) root two, (C) negative root two, (D) root two over two, or (E) negative root two over two.

The value 𝑥 is the quotient of two square roots. Before we consider the value of 𝑥 to the power of negative one, we can simplify the value of 𝑥 itself. Four is a square number, and its square root is equal to two. Hence, the value of 𝑥 is equal to two over root two. Now, this expression involves a radical in the denominator. And we may be concerned that we need to rationalize this denominator before we can proceed. But let’s first consider the expression we are asked to evaluate. It’s 𝑥 to the power of negative one.

We can recall the law for negative exponents, which states that for any nonzero real value of 𝑎, 𝑎 to the power of negative 𝑛 is equal to one over 𝑎 to the 𝑛th power. Hence, 𝑥 to the power of negative one is equal to one over 𝑥 to the first power, which is simply one over 𝑥. So what we’re actually being asked to find is the reciprocal of 𝑥. This means that the radical in the denominator of 𝑥 will be in the numerator of its reciprocal. So we don’t need to worry about rationalizing the denominator of 𝑥 before we can continue.

Substituting our simplified expression for 𝑥 gives one over 𝑥 is equal to one divided by two over root two. Dividing by a fraction is equivalent to multiplying by the reciprocal of that fraction. So one over 𝑥 is equal to one multiplied by root two over two, which is just equal to root two over two. We could also have got to this point by simply swapping the numerator and denominator of 𝑥 around to find its reciprocal. Looking at the five options given, the correct answer is option (D).

Now, if we had decided to rationalize the denominator in our expression for 𝑥 by multiplying both the numerator and denominator by root two, we would’ve obtained 𝑥 is equal to root two. Then, when evaluating one over 𝑥, we would’ve obtained one over root two. And the denominator would’ve required rationalizing again. Multiplying both the numerator and denominator by root two again would give root two over two, which is the same as our previous answer. So both methods give the same result.

Hence, by recalling the law for negative exponents and applying this to two equivalent expressions for 𝑥, we’ve found that if 𝑥 is equal to root four over root two, the value of 𝑥 to the power of negative one is root two over two.