What is the square root of 72 in simplest radical form?
Understand the Problem
The question is asking for the square root of 72 and specifically for it to be expressed in its simplest radical form. To solve this, we would first factor 72 into its prime factors and then simplify accordingly.
Answer
$6\sqrt{2}$
Answer for screen readers
The simplest radical form of the square root of 72 is $6\sqrt{2}$.
Steps to Solve
- Prime Factorization of 72
Start by breaking down 72 into its prime factors.
$$ 72 = 2^3 \times 3^2 $$
- Apply the Square Root Property
We can use the property of square roots that states $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$.
So,
$$ \sqrt{72} = \sqrt{2^3 \times 3^2} $$
- Separate the Perfect Squares
We will separate the factors into perfect squares and non-perfect squares.
We know that $2^2$ and $3^2$ are perfect squares:
$$ \sqrt{72} = \sqrt{(2^2) \times (2) \times (3^2)} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{2} $$
- Calculate the Square Roots of the Perfect Squares
Evaluate the square roots of the perfect squares:
$$ \sqrt{2^2} = 2 $$
$$ \sqrt{3^2} = 3 $$
- Combine the Results
Now we can combine our results:
$$ \sqrt{72} = 2 \times 3 \times \sqrt{2} = 6\sqrt{2} $$
The simplest radical form of the square root of 72 is $6\sqrt{2}$.
More Information
Expressing square roots in their simplest radical form helps in many areas of mathematics, including geometry, algebra, and more. The square root of 72 can also lead to approximations which can be useful in calculations.
Tips
- Forgetting to break down all the factors into their prime components, which can lead to incorrect roots.
- Not simplifying the square root fully by leaving out factors that are perfect squares.