What is the irrational square root √125 rewritten as
the product of an integer and another irrational
square root, in simplest form?
The irrational square root √125 can be rewritten as √(25*5). Since 25 is a perfect square, it can be simplified as 5. Thus, √125 can be written as 5√5.
To rewrite the irrational square root √125 as the product of an integer and another irrational square root, we need to express 125 as the product of a perfect square and another number.
First, let's find the largest perfect square that is less than or equal to 125. The perfect square that fits this criteria is 121 (11²).
Now, we can write 125 as the product of 121 and a remainder: 125 = 121 + 4.
Let's rewrite √125 using this expression:
√125 = √(121 + 4).
Now, we can split this under the square root into two separate terms:
√(121 + 4) = √121 + √4.
The perfect square root of 121 simplifies to 11, and the square root of 4 simplifies to 2.
So, √125 can be rewritten as:
√125 = √121 + √4
= 11 + 2√1.
Since √1 is equal to 1, we can simplify further:
√125 = 11 + 2(1)
= 11 + 2
= 13.
Therefore, the irrational square root √125 can be rewritten as the product of the integer 13 and another irrational square root, which is simply √1.
To rewrite √125 as the product of an integer and another irrational square root, we need to find the largest perfect square that divides 125. In this case, the largest perfect square that divides 125 is 25.
So, we can rewrite √125 as √(25 * 5).
Now, we can split the square root into two separate square roots as follows:
√(25 * 5) = √25 * √5
The square root of 25 is 5, which is an integer, and the square root of 5 is an irrational number that cannot be simplified further.
Therefore, √125 can be rewritten as 5√5, where 5 is an integer and √5 is an irrational square root.