(5a^2b^3)(6a^kb)=30a^6b^4

2 min read Jun 16, 2024

Solving for the Unknown Exponent

The equation (5a²b³)(6a^kb) = 30a⁶b⁴ presents a challenge: finding the value of k that makes the equation true. Let's break down the steps to solve this:

1. Simplify the Left Side

Multiply the coefficients: 5 * 6 = 30
Combine the 'a' terms: a² * a^k = a^(2+k)
Combine the 'b' terms: b³ * b = b^(3+1) = b⁴

Now, the simplified left side of the equation is 30a^(2+k)b⁴

2. Compare Coefficients and Exponents

The equation now reads: 30a^(2+k)b⁴ = 30a⁶b⁴

To make the equation true, both sides must have the same coefficients and exponents for each variable.

Coefficients: We already see that the coefficients (30) match on both sides.
Exponents: We need to ensure that the exponents for 'a' and 'b' match on both sides.

3. Solve for 'k'

'a' exponent: 2 + k = 6
Solve for 'k': k = 6 - 2 = 4

Conclusion

Therefore, the value of k that satisfies the equation (5a²b³)(6a^kb) = 30a⁶b⁴ is k = 4.