Multiply / Divide Radical Expressions.
What is the product of √6 and 9√12 in simplest radical form?
To find the product of √6 and 9√12, we multiply the coefficients and then multiply the radicands:
√6 * 9√12
= 9 * √6 * √12
Since √6 * √12 is equal to the square root of the product of the radicands, we have:
= 9 * √(6 * 12)
= 9 * √72
To simplify √72, we look for perfect square factors. The largest perfect square that divides 72 is 36, which is a factor of 72. Therefore:
√72 = √(36 * 2) = √36 * √2 = 6√2
Substituting this back into the expression, we get:
9 * 6√2 = 54√2
Therefore, the product of √6 and 9√12 in simplest radical form is 54√2.
To find the product of √6 and 9√12, we can multiply the coefficients (numbers outside the radical), as well as the radicands (numbers inside the radical).
The coefficient of √6 is 1, and the coefficient of 9√12 is 9. Multiplying the coefficients, we get 1 * 9 = 9.
The radicand of √6 is 6, and the radicand of 9√12 is 12. Multiplying the radicands, we get 6 * 12 = 72.
Therefore, the product of √6 and 9√12 is 9√72.
To simplify the radical, we can find the largest perfect square that divides evenly into 72. The largest perfect square that divides evenly into 72 is 36.
So, 72 can be written as 36 * 2.
Using the property of radicals, we can simplify 9√72 as 9√(36 * 2) = 9√36 * √2 = 9 * 6√2 = 54√2.
In simplest radical form, the product of √6 and 9√12 is 54√2.
To find the product of √6 and 9√12 in simplest radical form, follow these steps:
Step 1: Simplify each radical separately.
First, simplify √6:
√6 = √(2 * 3) = √2 * √3 = √2√3 = √6
Next, simplify 9√12:
9√12 = 9√(2 * 2 * 3) = 9√(2^2 * 3) = 9 * 2√3 = 18√3
Step 2: Multiply the simplified radicals.
Multiply √6 and 9√12:
√6 * 9√12 = √6 * 18√3 = 18√6√3
Step 3: Simplify the product, if possible.
Since both √6 and √3 are already simplified, we can simply multiply the coefficients:
18√6√3 = 18 * √6 * √3 = 18√18
Step 4: Further simplify, if possible.
To simplify 18√18, we need to find the largest perfect square that divides evenly into 18 and its exponent is 2. In this case, it is 9:
18√18 = 18√(9 * 2) = 18 * 3√2 = 54√2
Therefore, the product of √6 and 9√12 in simplest radical form is 54√2.