Problem 136 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$4^{-2}<4^{-3}$$

Step-by-Step Solution

Answer

The original given equation is false. After performing the required operations, the corrected true statement is \(x^3 - 64 = (x-4)(x^{2}+4x+16)\)

Step 1: Expansion of Right Side of the Equation

The first step involves the expansion of the right side of the equation which is basically a multiplication of a binomial with a trinomial. By using the distributive property, we expand this expression as follows: \((x+4)(x^{2}+4x-16)\) \(= x^2 * x + 4x * x + x* -16 + 4*x^{2} +4*4x +4*-16\) \(= x^3 + 4x^2 -16x +4x^2 -16x -64\)

Step 2: Simplification of Right Side of the Equation

Combine like terms in the resulting equation from step 1. \(= x^3 + 4x^2 + 4x^2 -16x -16x -64\) Hence, the simplified form of the right side of the equation is \(= x^3 + 8x^2 -32x -64\)

Step 3: Comparing the Left Side and the Right Side of the Equation

The left side of the given equation is \(x^3 - 64\). Upon comparing this with the simplified right side \(x^3 + 8x^2 -32x -64\), it’s evident that the two sides of the equation do not match. Therefore, the statement given in the problem is false.

Step 4: Make necessary changes to the right side of the equation

To make the original equation a true statement, the right side of the equation should be changed to match the left side. Thus, the correct equation would be \(x^3 - 64 = (x-4)(x^{2}+4x+16)\)