Question

Using properties of exponents which expression is equivalent to x/x^3/4

Answer 1

Options I was given
x
1/4 square root x
4 square root x
x ^4

Answer 2

To simplify the expression \( \frac{x}{x^{3/4}} \), we can use the properties of exponents. Specifically, we can use the rule that states:

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

Applying this rule to our expression:

\[
\frac{x}{x^{3/4}} = x^{1 - \frac{3}{4}}
\]

Now, simplify the exponent:

\[
1 - \frac{3}{4} = \frac{4}{4} - \frac{3}{4} = \frac{1}{4}
\]

So, we have:

\[
\frac{x}{x^{3/4}} = x^{1/4}
\]

Thus, the expression \( \frac{x}{x^{3/4}} \) is equivalent to:

\[
x^{1/4}
\]

This is the final answer.

Answer 3

Given the options, we can rewrite the expression we found, \( x^{1/4} \), to match one of the options.

The expression \( x^{1/4} \) can also be expressed as:

\[
x^{1/4} = \sqrt[4]{x}
\]

To see if this matches any of your options, let's evaluate each one:

1. **\( x \)**: This is \( x^1 \), which is not equivalent to \( x^{1/4} \).
2. **\( \frac{1}{4} \sqrt{x} \)**: This means \( \frac{1}{4} x^{1/2} \), which does not match \( x^{1/4} \).
3. **\( 4 \sqrt{x} \)**: This means \( 4 x^{1/2} \), which does not match \( x^{1/4} \).
4. **\( x^4 \)**: This is \( x^4 \), which is also not equivalent to \( x^{1/4} \).

None of the options exactly match \( x^{1/4} \) as it is or as \( \sqrt[4]{x} \).

However, it seems like the option **\( \frac{1}{4} \sqrt{x} \)** could potentially be a misinterpretation; if it is meant to point towards a fourth root but is directly referencing a fraction.

If no option directly aligns with \( x^{1/4} \) or \( \sqrt[4]{x} \), it appears that either there is a misunderstanding in the options or they do not correspond to the correct simplification of the original expression \( \frac{x}{x^{3/4}} \).

Let me know if you have more context or if those are the only options!