Simplify square root of (5-3)^2-(5-3)^2
The question is asking for the simplification of a mathematical expression involving square roots and exponents. It consists of two parts: the squared result of the expression (5-3), and then the subtraction of the same squared result of (5-3). The question tests understanding of basic algebraic manipulation and properties of square roots and exponents.
$\sqrt{\left(\left(\right. 5 - 3 \left.\right)\right)^{2} - \left(\left(\right. 5 - 3 \left.\right)\right)^{2}}$
Answer
Solution:
Step 1:
Calculate the difference between $5$ and $3$. $\sqrt{(5 - 3)^2 - (5 - 3)^2}$
Step 2:
Compute the square of $2$. $\sqrt{2^2 - (5 - 3)^2}$
Step 3:
Again, calculate the difference between $5$ and $3$. $\sqrt{4 - (5 - 3)^2}$
Step 4:
Compute the square of $2$. $\sqrt{4 - 2^2}$
Step 5:
Multiply $4$ by $-1$. $\sqrt{4 - 4}$
Step 6:
Perform the subtraction of $4$ from $4$. $\sqrt{0}$
Step 7:
Express $0$ as a square. $\sqrt{0^2}$
Step 8:
Extract terms from under the square root, assuming all are positive real numbers. $0$
Knowledge Notes:
The problem involves simplifying a mathematical expression that contains a square root and squares of a binomial. The key knowledge points include:
Order of Operations (PEMDAS/BODMAS): This is the sequence of steps used to solve mathematical expressions: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
Simplifying Square Roots: The square root of a number is a value that, when multiplied by itself, gives the original number. The square root of a perfect square (like $4$ or $9$) is an integer.
Squaring Binomials: When a binomial (a two-term algebraic expression) is squared, each term inside the binomial is squared, along with the cross-product of the terms, following the formula $(a - b)^2 = a^2 - 2ab + b^2$. However, in this case, the cross-product is not relevant because the same binomial is being subtracted, resulting in a cancellation.
Zero Property: The square root of zero is zero, and zero squared is also zero. This is because $0$ is the only number that, when multiplied by itself, results in $0$.
Simplification: The process of reducing an expression to its simplest form. In this case, the expression simplifies to zero because the terms under the square root cancel each other out.
Assumption of Positive Real Numbers: When simplifying square roots, it is often assumed that the result will be a positive real number unless otherwise specified, as the principal square root is always positive.
