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

1 answer

To express each of the given expressions with a single rational exponent, we will follow these steps:

1. Fourth root of \(x^3\)

The fourth root of a number can be expressed as that number raised to the power of \(1/4\). Thus, the fourth root of \(x^3\) can be expressed as: \[ \sqrt[4]{x^3} = (x^3)^{1/4} \] Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we get: \[ (x^3)^{1/4} = x^{3 \cdot \frac{1}{4}} = x^{\frac{3}{4}} \]

2. One over \(x^{-1}\)

This expression can be rewritten using the property of negative exponents, which states that \(a^{-n} = \frac{1}{a^n}\): \[ \frac{1}{x^{-1}} = x^{1} \]

3. Tenth root of the quantity \(x^5 \cdot x^4 \cdot x^2 \)

First, simplify the expression inside the parentheses. We can combine the exponents by adding them (since \(x^m \cdot x^n = x^{m+n}\)): \[ x^5 \cdot x^4 \cdot x^2 = x^{5 + 4 + 2} = x^{11} \] Now, we express the tenth root: \[ \sqrt[10]{x^{11}} = (x^{11})^{1/10} \] Using the power of a power rule again: \[ (x^{11})^{1/10} = x^{\frac{11}{10}} \]

4. \(x^{1/3} \cdot x^{1/3} \cdot x^{1/3}\)

Use the property of exponents (as shown previously) to combine the terms: \[ x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} = x^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = x^{\frac{1}{3} \cdot 3} = x^{1} \]

Summary of Expressions with Single Rational Exponents:

  1. Fourth root of \(x^3\) can be rewritten as:
    \[x^{\frac{3}{4}}\]

  2. One over \(x^{-1}\) can be rewritten as:
    \[x^{1}\]

  3. Tenth root of \(x^5 \cdot x^4 \cdot x^2\) can be rewritten as:
    \[x^{\frac{11}{10}}\]

  4. \(x^{1/3} \cdot x^{1/3} \cdot x^{1/3}\) can be rewritten as:
    \[x^{1}\]

Which Expressions are Equivalent?

Now we can identify which expressions are equivalent based on their final forms:

  • The expression \(x^{1}\) appears from both the second expression (\(1/x^{-1}\)) and the fourth expression (\(x^{1/3} \cdot x^{1/3} \cdot x^{1/3}\)).
  • The expressions \(x^{\frac{3}{4}}\) and \(x^{\frac{11}{10}}\) are different from each other and from \(x^1\).

Thus, the equivalent expressions are:

  • One over \(x^{-1}\) and \(x^{1/3} \cdot x^{1/3} \cdot x^{1/3}\) because both simplify to \(x\).