How can I write the expression with a rationalized denominator for sqrt(3) * sqrt(6) * sqrt(3) * sqrt(6)?

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To rationalize the denominator of the expression sqrt(3) * sqrt(6) * sqrt(3) * sqrt(6), we will follow a systematic approach. Here’s how:

Step 1: Multiply the Square Roots

First, we can simplify the expression by combining the square roots:

sqrt(3) * sqrt(6) * sqrt(3) * sqrt(6) = (sqrt(3) * sqrt(3)) * (sqrt(6) * sqrt(6))

This is equal to:

sqrt(3^2) * sqrt(6^2) = sqrt(9) * sqrt(36)

Step 2: Calculate the Square Roots

Now, calculate the square roots:

  • sqrt(9) = 3
  • sqrt(36) = 6

So, we have:

3 * 6 = 18

Step 3: Conclusion

The expression sqrt(3) * sqrt(6) * sqrt(3) * sqrt(6) simplifies down to 18.

In this particular case, the expression you started with has no denominator that needs rationalizing, as it is a whole number upon simplification. However, if you were dealing with a more complex fraction that included square roots in the denominator, you would multiply the numerator and denominator by the conjugate or the radical necessary to eliminate the root in the denominator.

For example, if your expression was something like:

1 / sqrt(3)

Then, you would rationalize it by multiplying both the numerator and denominator by sqrt(3), leading to:

(1 * sqrt(3)) / (sqrt(3) * sqrt(3)) = sqrt(3) / 3

This process ensures that the denominator is a rational number.

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