Question

Write each expression with a single rational exponent. Show each step of your process. Which expressions are equivalent? Justify your reasoning.


fourth root of x cubed
one over x to the negative one power
tenth root of the quantity x to the fifth times x to the fourth times x squared end quantity
x to the one third power times x to the one third power times x to the one third power

Answer 1

To rewrite each expression with a single rational exponent, we'll begin by recalling the definitions of rational exponents and roots:

- The \( n \)-th root of \( a \) can be written as \( a^{1/n} \).
- The expression \( a^{-n} \) is equivalent to \( \frac{1}{a^n} \).
- Products of the same base can be simplified using the rules of exponents.

Let's convert each expression step-by-step:

1. **Fourth root of \( x^3 \)**:
\[
\text{Fourth root of } x^3 = (x^3)^{1/4}
\]
Applying the exponent rule \( (a^m)^n = a^{m \cdot n} \):
\[
(x^3)^{1/4} = x^{3/4}
\]

2. **One over \( x \) to the negative one power**:
\[
\frac{1}{x^{-1}} = x^{1}
\]
(Since \( \frac{1}{x^{-1}} = x^1 \) by the property that \( a^{-n} = \frac{1}{a^n} \)).

3. **Tenth root of the quantity \( x^5 \cdot x^4 \cdot x^2 \)**:
First, simplify the expression inside the root:
\[
x^5 \cdot x^4 \cdot x^2 = x^{5+4+2} = x^{11}
\]
Now apply the tenth root:
\[
\text{Tenth root of } x^{11} = (x^{11})^{1/10} = x^{11/10}
\]

4. **\( x^{1/3} \) times \( x^{1/3} \) times \( x^{1/3} \)**:
\[
x^{1/3} \cdot x^{1/3} \cdot x^{1/3} = x^{1/3 + 1/3 + 1/3} = x^{3/3} = x^{1}
\]

Now we can summarize our results:
1. Fourth root of \( x^3 \) = \( x^{3/4} \)
2. One over \( x^{-1} \) = \( x^{1} \)
3. Tenth root of \( x^{11} \) = \( x^{11/10} \)
4. \( x^{1/3} \cdot x^{1/3} \cdot x^{1/3} \) = \( x^{1} \)

### Equivalent Expressions:
To determine which expressions are equivalent:

- **\( x^{3/4} \)** is not equivalent to any of the others.
- **\( x^{1} \)** appears in expressions 2 and 4, thus these two are equivalent.
- **\( x^{11/10} \)** is not equivalent to any of the others.

### Conclusion:
- The expressions equivalent to \( x^1 \) are **one over \( x^{-1} \)** and **\( x^{1/3} \times x^{1/3} \times x^{1/3} \)** since both evaluate to \( x^1 \).
- The expression \( x^{3/4} \) is unique and not equivalent to any other.
- The expression \( x^{11/10} \) is also unique and not equivalent to any other.