Sure! Let's simplify the given radical step-by-step.
### Step 1: Prime Factorization
First, we need to factorize [tex]\(243\)[/tex] into its prime factors.
[tex]\[243 = 3 \times 81\][/tex]
[tex]\[81 = 3 \times 27\][/tex]
[tex]\[27 = 3 \times 9\][/tex]
[tex]\[9 = 3 \times 3\][/tex]
So, the prime factorization of [tex]\(243\)[/tex] is:
[tex]\[243 = 3 \times 3 \times 3 \times 3 \times 3 = 3^5\][/tex]
### Step 2: Rewriting the Radical
Now, we rewrite [tex]\(\sqrt{243}\)[/tex] in terms of its prime factors:
[tex]\[\sqrt{243} = \sqrt{3^5}\][/tex]
### Step 3: Separating Into Perfect Squares
We can express [tex]\(3^5\)[/tex] as a product of squares:
[tex]\[\sqrt{3^5} = \sqrt{3^4 \cdot 3} = \sqrt{(3^2)^2 \cdot 3}\][/tex]
[tex]\[= \sqrt{(9)^2 \cdot 3}\][/tex]
### Step 4: Simplifying the Radical
The square root of a square can be simplified:
[tex]\[\sqrt{(9)^2 \cdot 3} = 9 \cdot \sqrt{3}\][/tex]
Therefore:
[tex]\[\sqrt{243} = 9 \sqrt{3}\][/tex]
### Step 5: Conclusion
So, [tex]\(\sqrt{243}\)[/tex] can be simplified to:
[tex]\[
\sqrt{243} = 9 \sqrt{3}
\][/tex]
### Numerical Verification
We can verify this result numerically:
[tex]\[
9 \times \sqrt{3} \approx 9 \times 1.732 = 15.588
\][/tex]
So, our simplified form [tex]\(9 \sqrt{3}\)[/tex] approximately equals [tex]\(15.588\)[/tex], confirming that our simplification is correct.
[tex]\[ \sqrt{243} = 9 \sqrt{3} \approx 15.588 \][/tex]
Thus, we have successfully reduced the radical.
