Question

take test: 05: exponential functions hw

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question 4

\\(3^a^b = 3^a 3^b\\)

true

false

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Explanation:

Step1: Recall exponent rules

The power of a power rule states that \((a^m)^n = a^{m\times n}\). For the left - hand side of the equation \([3^{a}]^{b}\), by the power of a power rule, we have \([3^{a}]^{b}=3^{a\times b}=3^{ab}\).

The product of powers rule states that \(a^m\times a^n=a^{m + n}\). For the right - hand side of the equation \(3^{a}\times3^{b}\), by the product of powers rule, we have \(3^{a}\times3^{b}=3^{a + b}\).

Step2: Compare the two sides

We have the left - hand side as \(3^{ab}\) and the right - hand side as \(3^{a + b}\). These two expressions are equal only when \(ab=a + b\) for all values of \(a\) and \(b\), which is not true (for example, if \(a = 2\) and \(b=3\), then \(ab = 6\) and \(a + b=5\), and \(3^{6}
eq3^{5}\)). So \([3^{a}]^{b}
eq3^{a}\times3^{b}\) in general.

Answer:

False