Simplifying The Square Root Of 128 Made Easy
Table of Contents :
To simplify the square root of 128, you first need to understand the basic principles of square roots and how to factor numbers efficiently. Let’s embark on this journey together to make the concept of square roots clearer and more accessible!
Understanding Square Roots
A square root of a number ( x ) is a value ( y ) such that ( y^2 = x ). In simpler terms, the square root of a number is a number which, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because ( 3 \times 3 = 9 ).
Breaking Down 128
The first step in simplifying the square root of 128 is to factor the number. Factorization helps us identify perfect squares that can be taken out of the square root.
Let's look at the prime factorization of 128:
- Start by dividing by 2 (the smallest prime number):
- ( 128 ÷ 2 = 64 )
- ( 64 ÷ 2 = 32 )
- ( 32 ÷ 2 = 16 )
- ( 16 ÷ 2 = 8 )
- ( 8 ÷ 2 = 4 )
- ( 4 ÷ 2 = 2 )
- ( 2 ÷ 2 = 1 )
Thus, the prime factorization of 128 can be written as:
[ 128 = 2^7 ]
Simplifying the Square Root
Now that we have the factorization, we can simplify the square root:
[ \sqrt{128} = \sqrt{2^7} ]
To simplify ( \sqrt{2^7} ), we can break it down:
[ \sqrt{2^7} = \sqrt{(2^6) \cdot (2^1)} = \sqrt{2^6} \cdot \sqrt{2^1} ]
Knowing that ( \sqrt{2^6} = 2^{6/2} = 2^3 = 8 ), we have:
[ \sqrt{2^7} = 8 \cdot \sqrt{2} ]
Thus, the simplified form of ( \sqrt{128} ) is:
[ \sqrt{128} = 8\sqrt{2} ]
Key Takeaways
- Factor the number: Finding the prime factorization of 128 was crucial in simplifying the square root.
- Recognize perfect squares: ( 2^6 ) was a perfect square, allowing us to simplify further.
- Express the result clearly: The final result of the square root of 128 is ( 8\sqrt{2} ).
Conclusion
In summary, the square root of 128 can be simplified to ( 8\sqrt{2} ). This process demonstrates how understanding the basics of square roots, prime factorization, and perfect squares can significantly simplify seemingly complex problems. Next time you encounter a square root, remember these steps and tackle the problem with confidence!