Let's simplify the expression [tex]\(\left(\frac{2}{3}\right)^2 \times \left(\frac{2}{3}\right)^3\)[/tex].
### Step-by-Step Solution:
1. Identify the base and exponents:
Both terms [tex]\(\left(\frac{2}{3}\right)^2\)[/tex] and [tex]\(\left(\frac{2}{3}\right)^3\)[/tex] share the same base, which is [tex]\(\frac{2}{3}\)[/tex].
2. Apply the laws of exponents:
When multiplying expressions with the same base, you add the exponents.
[tex]\[
\left(\frac{2}{3}\right)^2 \times \left(\frac{2}{3}\right)^3 = \left(\frac{2}{3}\right)^{2+3}
\][/tex]
3. Simplify the exponent:
Add the exponents:
[tex]\[
2 + 3 = 5
\][/tex]
So,
[tex]\[
\left(\frac{2}{3}\right)^{2+3} = \left(\frac{2}{3}\right)^5
\][/tex]
4. Calculate the results of the individual fractions:
Let's calculate each part separately:
[tex]\[
\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9} \approx 0.4444
\][/tex]
and
[tex]\[
\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27} \approx 0.2963
\][/tex]
5. Calculate the final simplified expression:
Raise [tex]\(\left(\frac{2}{3}\right)\)[/tex] to the power of 5:
[tex]\[
\left(\frac{2}{3}\right)^5 = \frac{2^5}{3^5} = \frac{32}{243} \approx 0.1317
\][/tex]
### Final Result:
- [tex]\(\left(\frac{2}{3}\right)^2 \approx 0.4444\)[/tex]
- [tex]\(\left(\frac{2}{3}\right)^3 \approx 0.2963\)[/tex]
- [tex]\(\left(\frac{2}{3}\right)^5 \approx 0.1317\)[/tex]
Thus, the simplified result of [tex]\(\left(\frac{2}{3}\right)^2 \times \left(\frac{2}{3}\right)^3\)[/tex] is approximately [tex]\(0.1317\)[/tex].
