Certainly! Let's solve the expression step by step:
Given:
[tex]\[
(9 \sqrt{8 x} - 1)(8 \sqrt{9 x} + 3)
\][/tex]
First, let's use the distributive property to expand the product of the two binomials.
Step 1: Distribute [tex]\(9 \sqrt{8 x}\)[/tex] across [tex]\(8 \sqrt{9 x} + 3\)[/tex]:
[tex]\[
9 \sqrt{8 x} \cdot 8 \sqrt{9 x} + 9 \sqrt{8 x} \cdot 3
\][/tex]
Step 2: Distribute [tex]\(-1\)[/tex] across [tex]\(8 \sqrt{9 x} + 3\)[/tex]:
[tex]\[
-1 \cdot 8 \sqrt{9 x} - 1 \cdot 3
\][/tex]
Putting these together, we have:
[tex]\[
(9 \sqrt{8 x})(8 \sqrt{9 x}) + (9 \sqrt{8 x})(3) + (-1)(8 \sqrt{9 x}) + (-1)(3)
\][/tex]
Let's simplify each term separately.
Term 1:
[tex]\[
(9 \sqrt{8 x})(8 \sqrt{9 x}) = 9 \cdot 8 \cdot \sqrt{8 x} \cdot \sqrt{9 x} = 72 \cdot \sqrt{72 x^2}
\][/tex]
Now, [tex]\(\sqrt{72 x^2}\)[/tex] can be simplified further:
[tex]\[
\sqrt{72 x^2} = \sqrt{36 \cdot 2 \cdot x^2} = 6 \sqrt{2} \cdot x
\][/tex]
So the first term becomes:
[tex]\[
72 \cdot 6 \sqrt{2} \cdot x = 432 \sqrt{2} \cdot x
\][/tex]
Term 2:
[tex]\[
(9 \sqrt{8 x})(3) = 27 \sqrt{8 x}
\][/tex]
We know that [tex]\(\sqrt{8} = 2 \sqrt{2}\)[/tex], therefore:
[tex]\[
27 \sqrt{8 x} = 27 \cdot 2 \sqrt{2 x} = 54 \sqrt{2} \sqrt{x} = 54 \sqrt{2 x}
\][/tex]
Term 3:
[tex]\[
-1 \cdot 8 \sqrt{9 x} = -8 \sqrt{9 x} = -8 \cdot 3 \sqrt{x} = -24 \sqrt{x}
\][/tex]
Term 4:
[tex]\[
-1 \cdot 3 = -3
\][/tex]
Putting all terms together, we get:
[tex]\[
432 \sqrt{2} x + 54 \sqrt{2 x} - 24 \sqrt{x} - 3
\][/tex]
Since [tex]\(\sqrt{2 x} = \sqrt{2} \sqrt{x}\)[/tex], we can rewrite [tex]\(54 \sqrt{2 x}\)[/tex] as [tex]\(54 \sqrt{2} \sqrt{x}\)[/tex].
Combining everything, we obtain:
[tex]\[
432 \sqrt{2} x + 54 \sqrt{2} \sqrt{x} - 24 \sqrt{x} - 3
\][/tex]
So the resulting expanded form is:
[tex]\[
-24\sqrt{x} + 54\sqrt{2}\sqrt{x} + 432\sqrt{2} x - 3
\][/tex]
That completes the detailed, step-by-step expansion of the given expression.
