Sure, let's simplify each expression step-by-step.
### Simplify: [tex]$\left(2^3\right)^3$[/tex]
Step 1: Identify the initial expression, which is [tex]$\left(2^3\right)^3$[/tex].
Step 2: Use the power of a power rule, which states [tex]$(a^m)^n = a^{m \times n}$[/tex].
Step 3: Apply the rule:
[tex]\[
(2^3)^3 = 2^{3 \times 3} = 2^9
\][/tex]
Thus, the simplified form of [tex]$\left(2^3\right)^3$[/tex] is [tex]$2^9$[/tex].
### Simplify: [tex]$\frac{5^{15}}{5^4}$[/tex]
Step 1: Identify the initial expression, which is [tex]$\frac{5^{15}}{5^4}$[/tex].
Step 2: Use the quotient of powers rule, which states [tex]$\frac{a^m}{a^n} = a^{m - n}$[/tex].
Step 3: Apply the rule:
[tex]\[
\frac{5^{15}}{5^4} = 5^{15 - 4} = 5^{11}
\][/tex]
Thus, the simplified form of [tex]$\frac{5^{15}}{5^4}$[/tex] is [tex]$5^{11}$[/tex].
### Verification with numerical results
For the expression [tex]$\left(2^3\right)^3$[/tex]:
[tex]\[
2^9 = 134217728
\][/tex]
For the expression [tex]$\frac{5^{15}}{5^4}$[/tex]:
[tex]\[
5^{11} = 48828125.0
\][/tex]
Your simplified expressions are in line with the numerical evaluations.
So, filling in the provided answers:
For [tex]$\left(2^3\right)^3$[/tex]:
- [tex]$2^9$[/tex]
For [tex]$\frac{5^{15}}{5^4}$[/tex]:
- [tex]$5^{11}$[/tex]
Ensure you check these boxes before submitting:
- [tex]$2^9$[/tex]
- [tex]$5^{11}$[/tex]
These simplified forms and their computed values are correct.
