To simplify the given expression
[tex]\[
(1+x)^{\frac{3}{2}} - (1+x)^{\frac{1}{2}},
\][/tex]
follow these steps:
1. Identify a common factor:
Notice that both terms share a common factor of [tex]\((1+x)^{\frac{1}{2}}\)[/tex]. We can factor this out from both terms.
2. Factor out the common term:
We can rewrite the expression as:
[tex]\[
(1+x)^{\frac{3}{2}} - (1+x)^{\frac{1}{2}} = (1+x)^{\frac{1}{2}} \left[ (1+x)^{\frac{3}{2} - \frac{1}{2}} - 1 \right].
\][/tex]
Here, we are factoring out [tex]\((1+x)^{\frac{1}{2}}\)[/tex].
3. Simplify the exponents inside the brackets:
Simplify the exponent inside the brackets:
[tex]\[
(1+x)^{\frac{3}{2} - \frac{1}{2}} = (1+x)^{\frac{3/2 - 1/2}} = (1+x)^1 = 1 + x.
\][/tex]
4. Subtract inside the brackets:
Now the expression inside the brackets simplifies to:
[tex]\[
(1+x)^1 - 1 = (1+x) - 1 = x.
\][/tex]
5. Combine the simplified terms:
Substituting back into the factored form, we get:
[tex]\[
(1+x)^{\frac{1}{2}} \cdot x.
\][/tex]
Hence, the simplified form of the expression
[tex]\[
(1+x)^{\frac{3}{2}} - (1+x)^{\frac{1}{2}}
\][/tex]
is
[tex]\[
(1+x)^{\frac{1}{2}} x.
\][/tex]
