Simplify

e^3ln2-1

Is the ln 2 part of the exponent?

If so, e^(3ln2) = (e^ln2)^3 = 2^3 = 8

Subtract 1 from that, for e^(3ln2) -1

To simplify the expression e^(3ln2 - 1), we can make use of two important logarithmic identities:

1. ln(a^b) = b * ln(a)
2. e^(ln(x)) = x

Using the first identity, we can rewrite the expression as:
e^(3ln2 - 1) = e^(ln2^3 - ln(e))

Since ln(e) is equal to 1 (as ln(e) is the natural logarithm of e, and e raised to the power of 1 equals e), the expression becomes:
e^(ln2^3 - ln(e)) = e^(ln2^3 - 1)

Next, using the second identity, we can simplify further:
e^(ln2^3 - 1) = 2^3 * e^(-1)

Simplifying the exponential term, we have:
2^3 * e^(-1) = 8 * e^(-1)

Therefore, the simplified form of the expression e^(3ln2 - 1) is 8 * e^(-1).