To simplify the given expressions, let's break down each of them step-by-step.
### Simplify: [tex]\((2^9)(2^2)\)[/tex]
1. Identify the bases and exponents:
- Here, [tex]\(2\)[/tex] is the base for both terms. The exponents are [tex]\(9\)[/tex] and [tex]\(2\)[/tex].
2. Apply the laws of exponents:
- When you multiply two exponents with the same base, you add the exponents. Mathematically, this is written as:
[tex]\[
a^m \cdot a^n = a^{m+n}
\][/tex]
3. Add the exponents:
- In our case, [tex]\(m = 9\)[/tex] and [tex]\(n = 2\)[/tex]:
[tex]\[
2^9 \cdot 2^2 = 2^{9+2} = 2^{11}
\][/tex]
### Simplify: [tex]\((2^3)^3\)[/tex]
1. Identify the base and the exponents:
- Again, [tex]\(2\)[/tex] is the base. The inner exponent is [tex]\(3\)[/tex], and we raise this to the power of [tex]\(3\)[/tex].
2. Apply the power rule of exponents:
- When you raise an exponent to another exponent, you multiply the exponents. Mathematically, this is written as:
[tex]\[
(a^m)^n = a^{m \cdot n}
\][/tex]
3. Multiply the exponents:
- In our case, [tex]\(m = 3\)[/tex] and [tex]\(n = 3\)[/tex]:
[tex]\[
(2^3)^3 = 2^{3 \cdot 3} = 2^9
\][/tex]
Thus, the simplified results are:
- For [tex]\((2^9)(2^2)\)[/tex]: [tex]\(2^{11}\)[/tex]
- For [tex]\((2^3)^3\)[/tex]: [tex]\(2^9\)[/tex]
Therefore, the answers are:
- [tex]\(\left(2^9\right)\left(2^2\right)\)[/tex]: [tex]\(2^{11}\)[/tex]
- [tex]\(\left(2^3\right)^3\)[/tex]: [tex]\(2^9\)[/tex]
