Let's start by using the prime factorization method to simplify the given expression:
[tex]\[
\frac{\sqrt[3]{125}}{\sqrt{11025}}
\][/tex]
### Step-by-Step Solution:
#### Step 1: Prime Factorization of 125
The number 125 can be factorized into prime factors:
[tex]\[
125 = 5 \times 5 \times 5 = 5^3
\][/tex]
#### Step 2: Cube Root of 125
Since [tex]\(125 = 5^3\)[/tex], taking the cube root of 125 gives:
[tex]\[
\sqrt[3]{125} = \sqrt[3]{5^3} = 5
\][/tex]
#### Step 3: Prime Factorization of 11025
Now let's factorize 11025:
[tex]\[
11025 = 105^2 = (3 \times 5 \times 7)^2
\][/tex]
Breaking this down further:
[tex]\[
11025 = 3^2 \times 5^2 \times 7^2
\][/tex]
#### Step 4: Square Root of 11025
Since [tex]\(11025 = (3 \times 5 \times 7)^2\)[/tex], taking the square root of 11025 gives:
[tex]\[
\sqrt{11025} = \sqrt{(3 \times 5 \times 7)^2} = 3 \times 5 \times 7 = 105
\][/tex]
#### Step 5: Forming the Fraction
With the cube root of 125 and the square root of 11025, we can form our fraction:
[tex]\[
\frac{\sqrt[3]{125}}{\sqrt{11025}} = \frac{5}{105}
\][/tex]
#### Step 6: Simplifying the Fraction
To simplify [tex]\(\frac{5}{105}\)[/tex], we find the greatest common divisor (GCD) of 5 and 105, which is 5. Dividing both the numerator and the denominator by their GCD:
[tex]\[
\frac{5 \div 5}{105 \div 5} = \frac{1}{21}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\frac{\sqrt[3]{125}}{\sqrt{11025}} = \frac{1}{21}
\][/tex]
So, the fraction [tex]$\frac{\sqrt[3]{125}}{\sqrt{11025}}$[/tex] simplified to its simplest form is:
[tex]\[
\frac{1}{21}
\][/tex]
