Show all of your steps for how to simplify 3^√5×2^√10
. Your final answer should be in simplified radical form (not a decimal). Answers only with no work shown will receive no credit.
To simplify the expression 3^√5 × 2^√10, we can follow these steps:
Step 1: Start by breaking down each of the numbers into their prime factors:
√5 = √(5) = √(5 × 1) = √5
√10 = √(2 × 5) = √(2 × 5 × 1) = √(10)
Step 2: Simplify each of the numbers to their simplest radical form:
√5 = √(5) = √5 (already in simplest radical form)
√10 = √(2 × 5) = √2 × √5 = √2√5
Step 3: Rewrite the expression using the simplified radical forms:
3^√5 × 2^√10 = 3^√5 × 2^(√2√5)
Step 4: Apply the exponent rules for multiplying exponents with the same base:
3^√5 × 2^(√2√5) = 3^√5 × 2^√2 × 2^√5
Step 5: Rewrite the expression using the simplified radical forms:
3^√5 × 2^√2 × 2^√5 = 3^√5 × 2^√2 × 2√5
Step 6: Combine the exponents:
3^√5 × 2^√2 × 2√5 = 3^√5 × 2^√2+√5
Step 7: Simplify the exponent:
2^√2+√5 = 2^(√2+√5)
Therefore, the simplified form of 3^√5 × 2^√10 is 3^√5 × 2^(√2+√5)
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.
To simplify the expression 3^√5 × 2^√10, we can first simplify each term separately, and then multiply them together.
Step 1: Simplify 3^√5
To simplify this term, we need to find a way to express the exponent as a rational number. We can achieve this by rationalizing the exponent.
√5 = (√5 × √5) / (√5 × √5)
= 5 / √25
= 5 / 5
= 1
Therefore, 3^√5 simplifies to 3^1, which is simply 3.
Step 2: Simplify 2^√10
Similar to the previous step, we want to rationalize the exponent.
√10 = (√10 × √10) / (√10 × √10)
= 10 / √100
= 10 / 10
= 1
Hence, 2^√10 simplifies to 2^1, which is just 2.
Step 3: Multiply the two simplified terms
Multiplying 3 and 2 together:
3 × 2 = 6
So, 3^√5 × 2^√10 simplifies to 6.
Therefore, the final answer in simplified radical form is 6.