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Chapter 0: Problem 129

Determine whether each statement makes sense or does not make sense, and explain your reasoning. There are many exponential expressions that are equal to \(36 x^{12},\) such as \(\left(6 x^{6}\right)^{2},\left(6 x^{3}\right)\left(6 x^{9}\right), 36\left(x^{3}\right)^{9},\) and \(6^{2}\left(x^{2}\right)^{6}\)

Short Answer

Expert verified
The statements \(\left(6 x^{6}\right)^{2}\), \(\left(6 x^{3}\right)\left(6 x^{9}\right)\), and \(6^{2}\left(x^{2}\right)^{6}\) makes sense and are equal to \(36x^{12}\), while the statement \(36\left(x^{3}\right)^{9}\) does not make sense since it is not equal to \(36x^{12}\).

Step by step solution

01

Analyze the \(1^{st}\) statement

First, let's look at the statement \( \left(6 x^{6}\right)^{2}\). According to the power of a power rule, this expression can be simplified by multiplying the exponents. Thus, \( \left(6 x^{6}\right)^{2}=36x^{12}\). So this statement makes sense.
02

Analyze the \(2^{nd}\) statement

Now let's look at the statement \( \left(6 x^{3}\right)\left(6 x^{9}\right)\). The multiplication rule for exponents tells us that when multiplying like bases, we add the exponents together. In this case, however, the exponents are applied to the variables separately from the coefficients in front. So, we have \( \left(6 * 6 * x^{(3+9)}\right) = 36x^{12}\). Therefore, this statement makes sense.
03

Analyze the \(3^{rd}\) statement

For the statement \( 36\left(x^{3}\right)^{9}\), applying the power of a power rule, we get \( 36x^{27}\). This is not equal to \(36x^{12}\), so this statement does not make sense.
04

Analyze the \(4^{th}\) statement

The last statement is \(6^{2}\left(x^{2}\right)^{6}\), which simplifies to \( 36x^{12}\) after applying the power of a power rule. Hence, this statement makes sense.

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