Alright, let's rewrite the given expressions in terms of radical expressions by making use of the properties of exponents and radicals.
First, let's simplify the rational expression:
[tex]$
\frac{y^{\frac{3}{4}}}{y^{\frac{1}{2}}}
$[/tex]
We start by using the properties of exponents. When dividing two expressions with the same base, we subtract the exponents:
[tex]$
y^{\frac{3}{4}} \div y^{\frac{1}{2}} = y^{\frac{3}{4} - \frac{1}{2}}
$[/tex]
To perform the subtraction, we need a common denominator. The common denominator for [tex]\(4\)[/tex] and [tex]\(2\)[/tex] is [tex]\(4\)[/tex]:
[tex]$
\frac{3}{4} - \frac{1}{2} = \frac{3}{4} - \frac{2}{4} = \frac{3 - 2}{4} = \frac{1}{4}
$[/tex]
So, the simplified expression is:
[tex]$
y^{\frac{1}{4}}
$[/tex]
Next, we convert [tex]\(y^{\frac{1}{4}}\)[/tex] to a radical expression. By definition, the exponent [tex]\( \frac{1}{4} \)[/tex] means we take the fourth root:
[tex]$
y^{\frac{1}{4}} = \sqrt[4]{y}
$[/tex]
For the second part, consider the expression:
[tex]$
\sqrt[8]{y^3}
$[/tex]
The radical expression [tex]\(\sqrt[8]{y^3}\)[/tex] can be rewritten using rational exponents. The [tex]\(8\)[/tex]th root of [tex]\(y^3\)[/tex] is the same as raising [tex]\(y^3\)[/tex] to the power of [tex]\(\frac{1}{8}\)[/tex]:
[tex]$
\sqrt[8]{y^3} = (y^3)^{\frac{1}{8}}
$[/tex]
We can apply the exponent rule [tex]\((a^m)^n = a^{mn}\)[/tex] here:
[tex]$
(y^3)^{\frac{1}{8}} = y^{3 \cdot \frac{1}{8}} = y^{\frac{3}{8}}
$[/tex]
In summary, our simplified expressions are:
1. The given rational exponent [tex]\(\frac{y^{\frac{3}{4}}}{y^{\frac{1}{2}}}\)[/tex] simplifies to [tex]\(y^{\frac{1}{4}}\)[/tex], which is equivalent to [tex]\(\sqrt[4]{y}\)[/tex].
2. The radical expression [tex]\(\sqrt[8]{y^3}\)[/tex] is the same as [tex]\((y^3)^{\frac{1}{8}}\)[/tex] and can be condensed as [tex]\(y^{\frac{3}{8}}\)[/tex].
Thus, the full answer, rephrased, is:
1. [tex]\(\frac{y^{\frac{3}{4}}}{y^{\frac{1}{2}}} = y^{\frac{1}{4}}\)[/tex], and as a radical expression, it is [tex]\(\sqrt[4]{y}\)[/tex].
2. [tex]\(\sqrt[8]{y^3} = (y^3)^{\frac{1}{8}} = y^{\frac{3}{8}}\)[/tex].
