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Chapter 2: Problem 27

Are \(3 t^{4}\) and \((3 t)^{4}\) equivalent? Why or why not?

Short Answer

Expert verified
No, \(3t^4\) and \((3t)^4\) are not equivalent because when we simplify the second expression, it becomes \(81t^4\), which has a different coefficient than the first expression.

Step by step solution

01

Simplify the first expression, \(3t^4\)

There is no further simplification needed for the first expression. We can leave it as \(3t^4\).
02

Simplify the second expression, \((3t)^4\)

To simplify the second expression, we need to apply the power of 4 to both 3 and \(t\) inside the parentheses. Using the rule \((ab)^n = a^n b^n\), we have: \((3t)^4 = 3^4t^4\) Now, calculate \(3^4\): \(3^4 = 3 \times 3 \times 3 \times 3 = 81\) So, \((3t)^4 = 81t^4\).
03

Compare the two expressions

Now, we have the following expressions: - First expression: \(3t^4\) - Second expression: \(81t^4\) We can see that \(3t^4\) and \(81t^4\) are not equivalent, as the coefficients 3 and 81 are different. So, \(3t^4\) and \((3t)^4\) are not equivalent expressions. To summarize, \(3t^4\) and \((3t)^4\) are not equivalent because when we simplify the second expression, it becomes \(81t^4\), which has a different coefficient than the first expression.

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