To rewrite the expression \(\sqrt{x} \times \sqrt[4]{x}\) in simplest rational exponent form, follow these steps:
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Rewrite the roots using rational exponents:
- The square root of \(x\) can be rewritten as \(x^{1/2}\).
- The fourth root of \(x\) can be rewritten as \(x^{1/4}\).
So we have: \[ \sqrt{x} = x^{1/2} \] \[ \sqrt[4]{x} = x^{1/4} \]
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Multiply the expressions: Now, substitute these forms back into the original expression: \[ \sqrt{x} \times \sqrt[4]{x} = x^{1/2} \times x^{1/4} \]
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Combine the exponents: When multiplying expressions with the same base, we add the exponents: \[ x^{1/2} \times x^{1/4} = x^{1/2 + 1/4} \]
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Find a common denominator: To add \(1/2\) and \(1/4\), we need a common denominator. The least common denominator of 2 and 4 is 4: \[ \frac{1}{2} = \frac{2}{4} \]
Now we can add: \[ \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \]
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Write the final expression: Substitute the sum back into the exponent: \[ x^{1/2 + 1/4} = x^{3/4} \]
Thus, the expression \(\sqrt{x} \times \sqrt[4]{x}\) in simplest rational exponent form is \(\boxed{x^{3/4}}\).
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