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Use the prime factorization method to work out the following, leaving your answer as a fraction in its simplest form.

[tex]\[
\frac{\sqrt[3]{125}}{\sqrt{11025}}
\][/tex]


Sagot :

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]
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