What is the product of 8, square root of, 8, end square root8 8 and 5, square root of, 24, end square root5 24 in simplest radical form
The Correct Answer and Explanation is:
To find the product of ( 8\sqrt{8} ) and ( 5\sqrt{24} ) in its simplest radical form, we can follow these steps:
Step 1: Break down the expression
The problem asks for the product of two numbers:
[
8\sqrt{8} \times 5\sqrt{24}
]
We can separate this into two parts: the constant multiplications and the radical parts:
[
(8 \times 5) \times (\sqrt{8} \times \sqrt{24})
]
Step 2: Multiply the constants
First, multiply the constants outside the square roots:
[
8 \times 5 = 40
]
Step 3: Multiply the square roots
Now, multiply the square roots. The product of two square roots can be written as a single square root:
[
\sqrt{8} \times \sqrt{24} = \sqrt{8 \times 24}
]
Simplify the multiplication inside the square root:
[
8 \times 24 = 192
]
Thus, we now have:
[
\sqrt{8} \times \sqrt{24} = \sqrt{192}
]
Step 4: Simplify the square root of 192
Next, simplify ( \sqrt{192} ). To do this, we factor ( 192 ) into its prime factors:
[
192 = 64 \times 3
]
We know that ( \sqrt{64} = 8 ), so:
[
\sqrt{192} = \sqrt{64 \times 3} = \sqrt{64} \times \sqrt{3} = 8\sqrt{3}
]
Step 5: Combine the results
Now, we multiply the constants and the simplified square root:
[
40 \times 8\sqrt{3} = 320\sqrt{3}
]
Final Answer
Therefore, the product of ( 8\sqrt{8} ) and ( 5\sqrt{24} ) in its simplest radical form is:
[
\boxed{320\sqrt{3}}
]