Multiply. Assume that all variables represent positive real numbers.

left parenthesis 3 x Superscript one half Baseline plus 4 right parenthesis left parenthesis 3 x Superscript one half Baseline minus 4 right parenthesis3x12+4 3x12−4

We are tasked with multiplying the following expression:

\[
(3x^{\frac{1}{2}} + 4)(3x^{\frac{1}{2}} - 4)
\]

This is a difference of squares, and it can be simplified using the identity:

\[
(a + b)(a - b) = a^2 - b^2
\]

Here, \( a = 3x^{\frac{1}{2}} \) and \( b = 4 \). Applying the difference of squares formula:

\[
(3x^{\frac{1}{2}} + 4)(3x^{\frac{1}{2}} - 4) = (3x^{\frac{1}{2}})^2 - 4^2
\]

Now, simplify each part:

\[
(3x^{\frac{1}{2}})^2 = 9x
\]
\[
4^2 = 16
\]

Thus, the expression becomes:

\[
9x - 16
\]

The final simplified expression is:

\[
9x - 16
\]

Would you like more details on any step, or do you have any further questions?

Here are 5 related questions for practice:
1. Expand and simplify: \( (5x^{\frac{1}{2}} + 6)(5x^{\frac{1}{2}} - 6) \)
2. Simplify \( (2x^{\frac{1}{2}} + 5)(2x^{\frac{1}{2}} - 5) \).
3. Factor \( x^2 - 49 \).
4. Solve for \( x \) in the equation \( (3x^{\frac{1}{2}} + 4)(3x^{\frac{1}{2}} - 4) = 0 \).
5. Simplify \( (7x^{\frac{1}{2}} + 2)(7x^{\frac{1}{2}} - 2) \).

**Tip**: When multiplying expressions that fit the form \( (a + b)(a - b) \), always remember to apply the difference of squares formula to simplify quickly.

Multiply. Assume that all variables represent positive real numbers.

(3x^{1/2} + 4)(3x^{1/2} - 4)

Learn how to multiply the expression (3x^{1/2} + 4)(3x^{1/2} - 4) using the difference of squares formula. This step-by-step solution covers algebraic concepts and simplifies the expression to 9x - 16. Suitable for Grades 9-11.

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