1) Is the equation: 28 - 4√2 = 24√2 True? Explain why or why not?
2) Is this statement √a + √b = √(a +b) True? Explain why or why not?
3) What is the index of a radical? When working with radicals, can the radicand be negative when the index is odd? Can it be negative when the index is even?
4) Simplify the following expressions:-
(a) 8√48 - 5√3
(b) 7 * 3√(-16) + 15 * 3√2
1. 28 - 4sqrt2 = 24sqrt2.
24sqrt2 + 4sqrt2 = 28,
28sqrt2 = 28. Not True.
Divide both sides by 28:
sqrt2 = 1. NOT TRUE.
2. sqrt(a) and sqrt(b) can be added only if a = b. So the Eq is Not True.
3. a. The index is 2 for a sqrt radical
and 3 for a crt radical.
b. Yes, the radican can be neg. when
the index is odd.
c. No, the radican cannot be neg. when the index is even.
4a. 8sqrt48 - 5sqrt3 =
8sqrt(16*3) - 5sqrt3 =
8*4sqrt3 - 5sqrt3 =
32sqrt3 - 5sqrt3 =
Factor out 5sqrt3:
sqrt3(32-5) = 27sqrt3.
b. 21sqrt(-16) + 45sqrt2 =
21sqrt(16*-1) + 45sqrt2 =
21*4sqrt(-1) + 45sqrt2 =
84sqrt(-1) + 45sqrt2 =
Factor out 3:
3(28sqrt(-1) + 15sqrt2 =
3(28i + 15sqrt2).
asd23erg
1) To determine if the equation is true, we can start by simplifying both sides of the equation and comparing them. Let's simplify the equation step by step:
First, simplify the left side:
28 - 4√2
Then, simplify the right side:
24√2
So, let's simplify the left side:
28 - 4√2 = 28 - 4 * √2 = 28 - 4√2
And now simplify the right side:
24√2 = 24 * √2 = 24√2
Now we'll compare the simplified left side with the simplified right side:
28 - 4√2 = 24√2
Since the two sides are not equal, the equation is not true. Therefore, the equation 28 - 4√2 = 24√2 is false.
2) To determine if the statement is true, we can use a similar approach:
Let's start by simplifying the left side of the equation:
√a + √b
Now, let's simplify the right side of the equation:
√(a + b)
These two expressions cannot be simplified further, so we can directly compare the two sides:
√a + √b = √(a + b)
Since the two sides are equal, the statement √a + √b = √(a + b) is true.
3) The index of a radical refers to the number outside the radical that indicates the root being taken. For example, in √x, the index is 2 because it represents a square root (2 is the root being taken).
When working with radicals, the radicand (the number inside the radical) can be negative only if the index is even. This is because an even root of a negative number will result in a positive number. For example, √(-4) is undefined because the square root of a negative number is not a real number. However, if the index were 4, ∜(-4) would be defined and equal to 2, since the fourth root of -4 is 2.
On the other hand, if the index is odd, the radicand cannot be negative, as taking an odd root of a negative number will result in a negative number. For instance, the cube root of -8 (∛(-8)) is -2, as the cube root of -8 is -2.
In summary, the radicand can be negative when the index is even, but it cannot be negative when the index is odd.
4)
(a) To simplify the expression 8√48 - 5√3, we can break down each term into its simplest form:
Start with √48:
√48 = √(16 * 3) = √16 * √3 = 4√3
Now, the expression becomes:
8 * 4√3 - 5√3
Simplify further:
32√3 - 5√3 = (32 - 5)√3 = 27√3
So, the simplified form of 8√48 - 5√3 is 27√3.
(b) To simplify the expression 7 * 3√(-16) + 15 * 3√2, we can apply the rules mentioned earlier:
Start with 3√(-16):
3√(-16) = 3 * √(-1 * 16) = 3 * √(-1) * √16 = 3 * i * 4 = 12i
Now, the expression becomes:
7 * 12i + 15 * 3√2
Simplify further:
84i + 45√2
The expression cannot be simplified any further since the terms involve different types of numbers (i, imaginary, and √2, irrational).
So, the simplified form of 7 * 3√(-16) + 15 * 3√2 is 84i + 45√2.