Answer :
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.
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.
