Express the product of 3¡Ì30(2¡Ì5-7) in simplest radical form.
30¡Ì6-21¡Ì30
To express the product of 3√30(2√5 - 7) in simplest radical form, we can follow these steps:
Step 1: Simplify the expression inside the parentheses.
2√5 - 7 cannot be simplified further because 2 and 7 are already in their simplest forms, and √5 cannot be simplified since 5 does not have any perfect square factors.
Step 2: Distribute the 3√30 to both terms inside the parentheses.
3√30(2√5 - 7) = 6√(30)(√5) - 21√(30)
Step 3: Simplify the radicals.
To simplify the radicals, we need to find any perfect square factors within the radicands (the numbers inside the square roots).
Let's break down 30 and 5 into their prime factors:
30 = 2 * 3 * 5
5 = 5
We can see that 5 is already a perfect square factor.
Step 4: Substitute the perfect square factors.
6√(30)(√5) - 21√(30) = 6√(2 * 3 * 5)(√5) - 21√(2 * 3 * 5)
Step 5: Simplify the radicals further.
We can now simplify the radicals by separating the square roots of the perfect square factors:
= 6√(2)(√3)(√5)(√5) - 21√(2)(√3)(√5)
Step 6: Simplify the perfect square factors.
Since we have two √5 terms here, we can simplify them as they are a perfect square:
= 6√(2)(√3)(5) - 21√(2)(√3)(5)
Step 7: Multiply the coefficients.
Now we can multiply the coefficients together:
= 30√(2)(√3) - 105√(2)(√3)
Step 8: Combine like terms.
Since both terms have the same radical parts, we can combine them:
= (30 - 105)√(2)(√3) = -75√(2)(√3)
Therefore, the expression 3√30(2√5 - 7) in its simplest radical form is -75√(2)(√3).