State whether the following are true or false.
The square roots of 36 are + and - 6. I wrote down false for this cause I thought it was only + 6, but the real answer is true, and I want to know that how can it be negative?
Also for this:
Convert the following radicals to mixed radicals in simplest form.
I did this without making the factor tree, my teacher showed me an easy way.
�ã50 = �ã2*5*5 = 5*5�ã2 = 25�ã2
That's what I got 25�ã2, but the answer is supposed to be 5�ã2. I don't understand, did they reduce or something?
Sorry this ã was supposed to be the radical sign, don't know why it never showed up.
A bit of a technicality here.
The √ symbol is defined as the principal square root, which means, the positive square root of a number
Thus √36 = +6
on the other hand if you have
x^2 = 16
then ±x = √16 = 4 from which we get
x = ± 4
so the -4 actually does not come from √16 but rather from
x^2 = 16
Your question is a poor choice for a true or false type question.
2nd part of your post:
√50 = √25√2 = 5√2
For the first question:
The statement "The square roots of 36 are + and - 6" is true. This is because the square root of a positive number has two solutions, one positive and one negative. In this case, the positive square root of 36 is indeed +6, and the negative square root of 36 is -6. The concept of square roots is based on the fact that when you square a number, the positive and negative square roots give you that number as a result. So, when you take the square root of 36, you need to consider both the positive and negative square roots.
For the second question:
To convert a radical to mixed radicals in simplest form, you need to factorize the number under the radical symbol into its prime factors and then simplify it. Let's go through the process for the example given, which is √50:
1. Find the prime factors of 50: 50 = 2 * 5 * 5.
2. Group the prime factors in pairs inside the radical so that one factor from each pair can come out as a whole number, like this: √(2 * 5 * 5) = √(5 * 5 * 2) = 5 * √2.
3. Simplify the expression: 5 * √2 is the simplest form of √50. This means that the radical cannot be further reduced.
In your attempt, you initially correctly factored 50 into 2 * 5 * 5. However, instead of grouping the prime factors in pairs, you only brought one 5 outside the radical, resulting in 25√2. This is not the simplest form because the 25 can be further simplified to 5^2. The correct answer is indeed 5√2, which is the simplest form.
So, to summarize: When converting a radical to mixed radicals in simplest form, factorize the number under the radical, group the prime factors in pairs, and simplify the expression by taking out any whole numbers that can be formed by multiplying identical factors together.