Certainly, let’s solve the problem step by step!
We start with the given expression:
[tex]$\frac{8^{30}}{8^3}$[/tex]
To simplify this expression, we can apply the properties of exponents. One such property is:
[tex]$\frac{a^m}{a^n} = a^{m-n}$[/tex]
Here, [tex]\(a\)[/tex] is the base (which is 8), [tex]\(m\)[/tex] is 30, and [tex]\(n\)[/tex] is 3. Applying the property, we get:
[tex]$\frac{8^{30}}{8^3} = 8^{30-3} = 8^{27}$[/tex]
Now, we need to determine which of the given choices is equivalent to [tex]\(8^{27}\)[/tex].
Let's evaluate the choices one by one:
1. [tex]\(\left(8^3\right)^9\)[/tex]:
Using the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]$\left(8^3\right)^9 = 8^{3 \cdot 9} = 8^{27}$[/tex]
This matches our simplified expression.
2. [tex]\(8^2 \cdot 8^5\)[/tex]:
Using the product rule [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]$8^2 \cdot 8^5 = 8^{2+5} = 8^7$[/tex]
This does not match [tex]\(8^{27}\)[/tex].
3. [tex]\(\left(8^1\right)^9\)[/tex]:
Again using the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]$\left(8^1\right)^9 = 8^{1 \cdot 9} = 8^9$[/tex]
This does not match [tex]\(8^{27}\)[/tex].
4. [tex]\(\frac{1}{8^{10}}\)[/tex]:
This can be rewritten using the negative exponent rule [tex]\(\frac{1}{a^n} = a^{-n}\)[/tex]:
[tex]$\frac{1}{8^{10}} = 8^{-10}$[/tex]
This does not match [tex]\(8^{27}\)[/tex].
Based on our evaluations, the expression equivalent to [tex]\(8^{27}\)[/tex] is:
[tex]$\left(8^3\right)^9$[/tex]
Therefore, the correct answer is:
[tex]$\boxed{\left(8^3\right)^9}$[/tex]
