Simplify the Radical Expression square root of 36+ square root of 64
The given problem is asking for the simplification of a mathematical expression that involves the square roots of two separate numbers, 36 and 64. The operation specified is addition, which means you need to find the square root of each individual number and then add them together. The term "simplify the radical expression" generally implies that the square roots should be calculated and expressed in the simplest form possible, meaning as whole numbers if the radicands (the numbers under the square root symbol) are perfect squares.
$\sqrt{36} + \sqrt{64}$
Answer
Solution:
Step:1
Break down each square root separately.
Step:1.1
Express $36$ as the square of $6$. $\sqrt{36} = \sqrt{6^2} + \sqrt{64}$
Step:1.2
Extract the square root of the perfect square. $6 + \sqrt{64}$
Step:1.3
Understanding absolute value as the non-negative value of a number. $6 + \sqrt{64}$
Step:1.4
Express $64$ as the square of $8$. $6 + \sqrt{8^2}$
Step:1.5
Extract the square root of the perfect square. $6 + 8$
Step:1.6
Recognize that the absolute value of a positive number is the number itself. $6 + 8$
Step:2
Combine the two numbers. $14$
Knowledge Notes:
The problem involves simplifying a radical expression that contains the square roots of two perfect squares, $36$ and $64$. The steps to simplify such an expression are as follows:
Identify Perfect Squares: Recognize that some numbers are perfect squares, meaning they are the square of an integer. In this case, $36$ is the square of $6$ ($6^2$), and $64$ is the square of $8$ ($8^2$).
Simplify Square Roots: Since the square root of a perfect square is the number that was squared, we can simplify $\sqrt{36}$ to $6$ and $\sqrt{64}$ to $8$.
Absolute Value: The absolute value of a number is its distance from zero on the number line, which is always non-negative. Since both $6$ and $8$ are non-negative, their absolute values are the same as the numbers themselves.
Arithmetic Operations: After simplifying the square roots, we perform any remaining arithmetic operations. In this case, we add the results to get the final answer.
Radical Notation: The radical symbol $\sqrt{}$ is used to denote the square root of a number. When simplifying expressions under a radical, it's important to consider whether the result should be positive or negative. Since square roots are defined to be non-negative, the result should be the positive root.
Combining Like Terms: After simplifying the individual terms, we combine them if possible. Here, we are adding two integers.
Final Simplification: The last step is to simplify the expression to its simplest form, which may involve adding, subtracting, multiplying, or dividing numbers.
Understanding these concepts is essential for simplifying radical expressions and performing operations with square roots.
