To simplify the expression [tex]\(\frac{3}{4} x^{1/2} \div \frac{1}{2} x^{-1/4}\)[/tex], we will follow these steps:
1. Rewrite the division as multiplication by the reciprocal:
[tex]\[
\frac{3}{4} x^{1/2} \div \frac{1}{2} x^{-1/4} = \frac{3}{4} x^{1/2} \times \frac{2}{1} x^{1/4}
\][/tex]
2. Simplify the constant coefficients:
[tex]\[
\frac{3}{4} \times \frac{2}{1} = \frac{3 \cdot 2}{4 \cdot 1} = \frac{6}{4} = \frac{3}{2}
\][/tex]
3. Combine the exponents using the properties of exponents:
[tex]\[
x^{1/2} \times x^{-1/4} = x^{1/2 + (-1/4)} = x^{1/2 - 1/4}
\][/tex]
4. Simplify the exponent:
[tex]\[
1/2 - 1/4 = \frac{2}{4} - \frac{1}{4} = \frac{1}{4}
\][/tex]
So,
[tex]\[
x^{1/2 - 1/4} = x^{1/4}
\][/tex]
Combining the simplified constant and the simplified variable, the final, simplified expression is:
[tex]\[
\frac{3}{2} x^{1/4}
\][/tex]
