Verified Answer :

  • Multiply coefficients and combine exponents using the rule \(x^m \cdot x^n = x^{m+n}\).
  • Simplify the left side to \(30a^{2+k}b^4\).
  • Equate the exponent of \(a\) on the left to the exponent of \(a\) on the right: \(2+k=6\).
  • Solve the linear equation \(2+k=6\) to find \(k = \boxed{4}\).

Explanation

  1. Problem Analysis and Data Review

We are given the equation \((5a^2b^3)(6a^kb) = 30a^6b^4\). Our goal is to determine the unknown value of the exponent \(k\). The base variables are \(a\) and \(b\), and the equation involves multiplying monomials. The product of two monomials is found by multiplying their coefficients and applying the laws of exponents to each variable base. The fundamental law of exponents we need is \(x^m \cdot x^n = x^{m+n}\).

  1. Simplifying the Equation

First, we multiply the coefficients on the left side of the equation: \(5 \times 6 = 30\). Next, we group the terms with the same base: \((5 \times 6) \cdot (a^2 \cdot a^k) \cdot (b^3 \cdot b^1)\). By applying the law of exponents \(x^m \cdot x^n = x^{m+n}\), we combine the terms for each base:

  • For \(a\): \(a^2 \cdot a^k = a^{2+k}\).
  • For \(b\): \(b^3 \cdot b^1 = b^{3+1} = b^4\).

This simplifies the left side of the equation to \(30 \cdot a^{2+k} \cdot b^4\), or \(30a^{2+k}b^4\).

  1. Comparing Exponents

Now, we compare the simplified left side, \(30a^{2+k}b^4\), with the right side of the original equation, \(30a^6b^4\). For the two expressions to be equal, the exponents of the corresponding bases must be equal. Comparing the exponents of base \(a\), we obtain the equation: \(2 + k = 6\). Comparing the exponents of base \(b\), we see that \(4 = 4\), which confirms our simplification is correct.

  1. Solving for k

Finally, we solve the linear equation \(2 + k = 6\) for \(k\):
\[k = 6 - 2\]
\[k = 4\]
Thus, the value of \(k\) that makes the original equation true is \(4\). Looking at the options provided, this corresponds to option (d).

Examples

Imagine you are managing a small warehouse where items are packaged in bulk. If you need to combine two shipments of products, where the first shipment consists of 5 units of product type 'a' squared and 'b' cubed, and the second shipment has 6 units of product type 'a' to the power of 'k' and 'b' to the power of 1, the resulting total inventory is given by multiplying these. If your final inventory counts show exactly 30 units of 'a' to the 6th power and 'b' to the 4th power, you can use the laws of exponents to determine the unknown quantity 'k' in your second shipment batch. This ensures your stock counts are accurate for inventory management and supply chain planning.

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