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Chapter 0: Problem 136

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

Short Answer

Expert verified
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 by step solution

01

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: \((x+4)(x^{2}+4x-16)\) \(= x^2 * x + 4x * x + x* -16 + 4*x^{2} +4*4x +4*-16\) \(= x^3 + 4x^2 -16x +4x^2 -16x -64\)
02

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 \(= x^3 + 8x^2 -32x -64\)
03

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鈥檚 evident that the two sides of the equation do not match. Therefore, the statement given in the problem is false.
04

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 \(x^3 - 64 = (x-4)(x^{2}+4x+16)\)

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