Square root of 112 in radical form
Understand the Problem
The question is requesting the square root of 112 to be expressed in radical form. To solve this, we will simplify the square root by breaking down 112 into its prime factors and extracting any perfect squares.
Answer
The square root of 112 in radical form is \( 4\sqrt{7} \).
Answer for screen readers
The square root of 112 in radical form is ( 4\sqrt{7} ).
Steps to Solve
- Find the prime factorization of 112
We start by breaking down 112 into its prime factors.
112 can be divided by 2: $$ 112 \div 2 = 56 $$
Continue factoring 56: $$ 56 \div 2 = 28 $$
Now, factor 28: $$ 28 \div 2 = 14 $$
Finally, factor 14: $$ 14 \div 2 = 7 $$
So, the prime factorization of 112 is: $$ 112 = 2^4 \cdot 7 $$
- Identify the perfect squares
From the factorization, we can identify that (2^4) is a perfect square. Perfect squares are numbers that have integer square roots.
In this case, we can rewrite (2^4) as:
$$ (2^2)^2 = 4^2 $$
- Simplify the square root
Now we can rewrite the square root of 112 using the perfect square:
$$ \sqrt{112} = \sqrt{(2^4) \cdot 7} $$
This can be further simplified using the property of square roots:
$$ \sqrt{112} = \sqrt{(2^4)} \cdot \sqrt{7} = 4\sqrt{7} $$
Thus, the square root of 112 in radical form is:
$$ \sqrt{112} = 4\sqrt{7} $$
The square root of 112 in radical form is ( 4\sqrt{7} ).
More Information
Expressing numbers in radical form can simplify calculations in math, particularly in algebra. The simplification process helps in identifying perfect squares and makes calculations easier in further operations.
Tips
- Not fully breaking down the number to its prime factors, missing any factors can lead to an incorrect answer.
- Failing to recognize perfect squares can also result in not simplifying the radical fully.
