Let's solve the given algebraic equation step by step to determine the value of [tex]\( k \)[/tex] that makes the equation true.
Given equation:
[tex]\[
\left(5 a^2 b^3\right)\left(8 a^4 b\right) = 30 a^6 b^4
\][/tex]
First, we'll simplify the left-hand side of the equation:
[tex]\[
\left(5 a^2 b^3\right)\left(8 a^4 b\right)
\][/tex]
Combine the numerical coefficients:
[tex]\[
5 \times 8 = 40
\][/tex]
Now, combine the powers of [tex]\( a \)[/tex]:
[tex]\[
a^2 \times a^4 = a^{2+4} = a^6
\][/tex]
And combine the powers of [tex]\( b \)[/tex]:
[tex]\[
b^3 \times b = b^{3+1} = b^4
\][/tex]
So, the left-hand side simplifies to:
[tex]\[
40 a^6 b^4
\][/tex]
Therefore, the equation becomes:
[tex]\[
40 a^6 b^4 = 30 a^6 b^4
\][/tex]
Now, let's compare both sides:
[tex]\[
40 a^6 b^4 \neq 30 a^6 b^4
\][/tex]
Since the coefficients [tex]\( 40 \)[/tex] and [tex]\( 30 \)[/tex] are not equal, the given equation is not true for any value of [tex]\( k \)[/tex].
None of the given values (2, 3, 4, 8) can make the equation true because the left-hand side simplifies to a different numerical coefficient than the right-hand side.
Therefore, there is no value of [tex]\( k \)[/tex] that makes the equation true. Hence, the correct answer is:
None of the values
