Answer :

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]