Show all of your steps for how to simplify 3^√5×2^√10

Question

. Your final answer should be in simplified radical form (not a decimal). Answers only with no work shown will receive no credit.

Answer 1

To simplify the expression 3^√5 × 2^√10, we can start by breaking down each term.

Let's start with 3^√5:
To simplify this term, we need to break down the exponent √5 into two factors. We can rewrite √5 as √(√(2^2 × 5)). This can be further simplified as √(2^2) × √5, which is 2 × √5. Therefore, 3^√5 = 3^(2 × √5) can be rewritten as 3^2 × 3^√5 = 9 × 2 × √5.

Now let's simplify 2^√10:
Similarly, we can break down the exponent √10 into two factors. We can rewrite √10 as √(√(2 × 5)). This can be further simplified as √(2) × √(5), which is √2 × √5. Therefore, 2^√10 = 2^(√2 × √5) can be rewritten as 2^(√2) × 2^√5 = √(2) × 2 × √5.

Now that we have simplified each term separately, we can multiply them together:
(9 × 2 × √5) × (√(2) × 2 × √5)

To multiply these terms, we can simply multiply the numbers and combine the square roots:
(9 × 2 × 2) × (√5 × √(2) × √5)

This simplifies to:
36 × √(5 × 2 × 5)

Further simplifying:
36 × √(5 × 2) × √5
36 × √10 × √5

Finally, combining the square roots and multiplying the numbers:
36√(10 × 5) = 36√50

Since 50 can be simplified as 25 × 2, we can rewrite the expression as:
36√(25 × 2)

And simplifying further:
36 × 5√2 = 180√2

Therefore, the simplified form of 3^√5 × 2^√10 is 180√2.