Question

Using the properties of exponents, which expression is equivalent to (z^(1/2)) / z^(1/2)?

The answer options are:

sqrt(z)

1/sqrt(z)

z^4

x

Answer 1

To simplify the expression \((z^{1/2}) / z^{1/2}\) using the properties of exponents, we can apply the quotient property of exponents, which states that:

\[
\frac{a^m}{a^n} = a^{m-n}
\]

In this case, both the numerator and denominator have the same base \(z\) with the exponent \(1/2\). Therefore, we can rewrite the expression as follows:

\[
(z^{1/2}) / z^{1/2} = z^{1/2 - 1/2} = z^{0}
\]

According to the zero exponent rule, any non-zero number raised to the power of 0 is equal to 1:

\[
z^{0} = 1
\]

Thus, \((z^{1/2}) / z^{1/2} = 1\).

Now looking at the options you provided:
1. \(\sqrt{z}\)
2. \(1/\sqrt{z}\)
3. \(z^4\)
4. \(x\)

None of these options represent the value we derived which is 1. The original expression is simply equal to 1, which isn't listed in your answer choices provided. If there's any further detail or context that needs to be considered, please let me know!