To factor the expression [tex]\( 2 x^{\frac{1}{4}} + 20 x^{\frac{1}{2}} \)[/tex], follow these steps:
1. Identify the common factor in the terms of the expression.
The expression is:
[tex]\[
2 x^{\frac{1}{4}} + 20 x^{\frac{1}{2}}
\][/tex]
2. Factor out the greatest common factor from the expression.
Both terms share a common factor of [tex]\( 2 \)[/tex]. Let’s factor the [tex]\( 2 \)[/tex] out:
[tex]\[
2 ( x^{\frac{1}{4}} + 10 x^{\frac{1}{2}} )
\][/tex]
3. Simplify the remaining expression inside the parentheses.
Notice that within the parentheses, each term involves [tex]\( x \)[/tex] raised to a power. To factor further if possible, we should express the terms using the same base power of [tex]\( x \)[/tex]:
[tex]\[
2 \left( x^{\frac{1}{4}} + 10 x^{\frac{2}{4}} \right)
\][/tex]
Here, [tex]\( x^{\frac{1}{2}} \)[/tex] is rewritten as [tex]\( x^{\frac{2}{4}} \)[/tex].
4. Compare and identify common factors inside the parentheses (Optional if needed for complex factors).
Given the simplified terms are now [tex]\( x^{\frac{1}{4}} \)[/tex] and [tex]\( x^{\frac{2}{4}} \)[/tex] (already simplified):
[tex]\[
2 \left( x^{0.25} + 10 x^{0.5} \right)
\][/tex]
Since there are no further common factors within the parentheses, the expression remains:
[tex]\[
2 ( x^{0.25} + 10 x^{0.5} )
\][/tex]
Thus, the final factored form of the given expression [tex]\( 2 x^{\frac{1}{4}} + 20 x^{\frac{1}{2}} \)[/tex] is:
[tex]\[
2 ( x^{\frac{1}{4}} + 10 x^{\frac{1}{2}} )
\][/tex]
Or equivalently:
[tex]\[
2 ( x^{0.25} + 10 x^{0.5} )
\][/tex]
