Question

Write a convincing argument to show that the power of a product property is true.

Short answer

The power of a product property is true, as shown by a proof using mathematical induction. Once you've established a base case (e.g., \(n=2\)), you assume the rule holds for \(n=k\) and then prove it for \(n=k+1\). When you show this progression, you've proved the rule holds for all integers.

Step-by-step solution

Understand the problem

The task is to provide a convincing proof of the power of a product property which states that the power of a product of two numbers equals the product of each number to that power separately.

Use reasoning

This property can be proven by reasoning on a finite number of cases, using an exponent of 2 for simplicity: taking (ab)^2 = a^2 * b^2. Expanding the left-hand side results in a*a*b*b and the right-hand side results in a*a*b*b as well.

Demonstrate induction

For a more general proof though, one can use mathematical induction. Assuming the property holds true for a certain \(k\), or \((ab)^k = a^k * b^k\). The crucial step is to show that it holds true for \(k + 1\). To do this, multiply both sides of the assumed equation by \(ab\).

Finalize the proof

From step 3, it follows that \((ab)^k * ab = a^k * b^k * ab\). This simplifies to \((ab)^{k+1} = a^k * a * b^k * b = a^{k+1} * b^{k+1}\). Therefore, if the property is true for \(k\), it is also true for \(k + 1\). Since we have already proven it true for 2, or any other base case, it is true for all integers by mathematical induction.