Sure! Let's work through this step-by-step.
### Volume Comparison of the Planters
1. Volume of Tree Planter:
The tree planter has a volume given by [tex]\(2^7\)[/tex] cubic feet. We know that:
[tex]\[
2^7 = 128 \text{ cubic feet}
\][/tex]
2. Volume of Shrub Planter:
The shrub planter has a volume given by [tex]\(2^4\)[/tex] cubic feet. We know that:
[tex]\[
2^4 = 16 \text{ cubic feet}
\][/tex]
3. Ratio of Volumes:
To determine how many times the volume of the tree planter is compared to the shrub planter, we divide the volume of the tree planter by the volume of the shrub planter:
[tex]\[
\frac{128 \text{ cubic feet}}{16 \text{ cubic feet}} = 8
\][/tex]
Therefore, the tree planter has 8 times the volume of the shrub planter.
### Simplifying the Given Expression
1. Given Expression:
Simplify the expression:
[tex]\[
\frac{2^3 \cdot 5^6 \cdot 9^5}{2^2 \cdot 5^3 \cdot 9^2}
\][/tex]
2. Break Down and Simplify Step-by-Step:
Let's handle each base separately.
- For the base 2:
[tex]\[
\frac{2^3}{2^2} = 2^{3-2} = 2^1 = 2
\][/tex]
- For the base 5:
[tex]\[
\frac{5^6}{5^3} = 5^{6-3} = 5^3
\][/tex]
- For the base 9:
Recall that [tex]\(9 = 3^2\)[/tex], so:
[tex]\[
9^5 = (3^2)^5 = 3^{10}
\][/tex]
and
[tex]\[
9^2 = (3^2)^2 = 3^4
\][/tex]
Therefore:
[tex]\[
\frac{9^5}{9^2} = \frac{3^{10}}{3^4} = 3^{10-4} = 3^6
\][/tex]
But, for simplicity, let's work directly with 9:
[tex]\[
\frac{9^5}{9^2} = 9^{5-2} = 9^3
\][/tex]
3. Combine the Results:
Putting it all together:
[tex]\[
2 \cdot 5^3 \cdot 9^3
\][/tex]
4. Calculate the Final Value:
We now simplify further:
- [tex]\( 5^3 = 125 \)[/tex]
- [tex]\( 9^3 = 729 \)[/tex]
Then, multiplying these values together:
[tex]\[
2 \cdot 125 \cdot 729 = 2 \cdot 91225 = 182250
\][/tex]
Therefore, the simplified form of the expression is:
[tex]\[
182250
\][/tex]
So, combining the results, we find:
- The tree planter has 8 times the volume of the shrub planter.
- The simplified expression [tex]\(\frac{2^3 \cdot 5^6 \cdot 9^5}{2^2 \cdot 5^3 \cdot 9^2}\)[/tex] equals [tex]\(182250\)[/tex].
