To determine the value of \( k \) that makes the equation \((5a^2 b^3)(6a^k b) = 30a^6 b^4\) true, let's break it down step-by-step.
Given equation:
[tex]\[
(5a^2 b^3)(6a^k b) = 30a^6 b^4
\][/tex]
1. Combine the numerical coefficients:
[tex]\[
5 \times 6 = 30
\][/tex]
2. Combine the powers of \( a \):
The product on the left side involving \( a \) is:
[tex]\[
a^2 \times a^k = a^{2+k}
\][/tex]
The right side of the equation has \( a^6 \).
Therefore, we set the exponents of \( a \) equal to each other:
[tex]\[
2 + k = 6
\][/tex]
Solving for \( k \):
[tex]\[
k = 6 - 2
\][/tex]
[tex]\[
k = 4
\][/tex]
3. Combine the powers of \( b \):
The product on the left side involving \( b \) is:
[tex]\[
b^3 \times b = b^{3+1} = b^4
\][/tex]
The right side of the equation also has \( b^4 \), so this matches perfectly.
Thus, all parts of the equation balance correctly when \( k \) is:
[tex]\[
k = 4
\][/tex]
Hence, the value of [tex]\( k \)[/tex] that makes the equation true is 4.
