Question

use the quotient rule to find the simplify the expression
8
---
8^(2/3)

Answer 1

To simplify the expression \(\frac{8}{8^{2/3}}\) using the properties of exponents, we can apply the quotient rule for exponents, which states that

\[
\frac{a^m}{a^n} = a^{m-n}.
\]

Here, we have \(a = 8\), \(m = 1\) (since \(8\) can be written as \(8^1\)), and \(n = \frac{2}{3}\).

Using the quotient rule:

\[
\frac{8}{8^{2/3}} = \frac{8^1}{8^{2/3}} = 8^{1 - \frac{2}{3}}.
\]

Now we calculate the exponent:

\[
1 - \frac{2}{3} = \frac{3}{3} - \frac{2}{3} = \frac{1}{3}.
\]

Thus, we have:

\[
8^{1 - \frac{2}{3}} = 8^{\frac{1}{3}}.
\]

Next, we can evaluate \(8^{\frac{1}{3}}\):

The cube root of \(8\) is \(2\), because \(2^3 = 8\).

Therefore, the simplified expression is:

\[
\frac{8}{8^{2/3}} = 2.
\]