To solve for [tex]\( \sqrt{3a - 8b} \)[/tex] given the expressions:
[tex]\[
\begin{aligned}
\text{M(x; y)} &= -8 x^{a-b} \cdot y^{a+2b}, \\
\text{N(x; y)} &= 21 x^7 \cdot y^{18-a},
\end{aligned}
\][/tex]
we need to determine the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]. For the terms to be similar, the powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in both expressions must be equal. Therefore, we can set up the following system of equations based on the exponents:
1. From the exponents of [tex]\(x\)[/tex]:
[tex]\[
a - b = 7
\][/tex]
2. From the exponents of [tex]\(y\)[/tex]:
[tex]\[
a + 2b = 18 - a
\][/tex]
First, rewrite the second equation to make it easier to solve:
[tex]\[
a + 2b + a = 18
\][/tex]
[tex]\[
2a + 2b = 18
\][/tex]
[tex]\[
a + b = 9
\][/tex]
We now have the system of linear equations:
[tex]\[
\begin{cases}
a - b = 7 \\
a + b = 9
\end{cases}
\][/tex]
We can solve this system using substitution or elimination. Here, we will use the elimination method. Add the two equations to eliminate [tex]\(b\)[/tex]:
[tex]\[
(a - b) + (a + b) = 7 + 9
\][/tex]
[tex]\[
2a = 16
\][/tex]
[tex]\[
a = 8
\][/tex]
Now substitute [tex]\(a = 8\)[/tex] into one of the original equations to find [tex]\(b\)[/tex]. We'll use [tex]\(a + b = 9\)[/tex]:
[tex]\[
8 + b = 9
\][/tex]
[tex]\[
b = 1
\][/tex]
So, the values are [tex]\(a = 8\)[/tex] and [tex]\(b = 1\)[/tex].
Next, we need to calculate [tex]\( \sqrt{3a - 8b} \)[/tex]:
[tex]\[
\sqrt{3a - 8b} = \sqrt{3(8) - 8(1)} = \sqrt{24 - 8} = \sqrt{16} = 4
\][/tex]
Thus, the final value is:
[tex]\[
\boxed{4}
\][/tex]
