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:
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Fourth root of \( x^3 \): \[ \text{Fourth root of } x^3 = (x^3)^{1/4} \] Applying the exponent rule \( (a^m)^n = a^{m \cdot n} \): \[ (x^3)^{1/4} = x^{3/4} \]
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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 \( a^{-n} = \frac{1}{a^n} \)).
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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: \[ \text{Tenth root of } x^{11} = (x^{11})^{1/10} = x^{11/10} \]
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\( x^{1/3} \) 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:
- Fourth root of \( x^3 \) = \( x^{3/4} \)
- One over \( x^{-1} \) = \( x^{1} \)
- Tenth root of \( x^{11} \) = \( x^{11/10} \)
- \( x^{1/3} \cdot x^{1/3} \cdot x^{1/3} \) = \( x^{1} \)
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.