Problem 33
Question
Evaluate the expression. $$ \sqrt{3}|-2|+3|-\sqrt{3}| $$
Step-by-Step Solution
Verified Answer
The given expression can be simplified and evaluated as follows: \(\sqrt{3}|-2|+3|-\sqrt{3}|\) becomes \(\sqrt{3}(2) + 3(\sqrt{3})\), which further simplifies to \(2\sqrt{3} + 3\sqrt{3}\), and finally evaluates to \(5\sqrt{3}\).
1Step 1: Identify and calculate the absolute values
Begin by finding the absolute values in the expression: |-2| and |-\sqrt{3}|.
We know that the absolute value of a number is always non-negative.
|-2| = 2 (since -2 is 2 units away from 0)
|-\sqrt{3}| = \(\sqrt{3}\) (since \(-\sqrt{3}\) is \(\sqrt{3}\) units away from 0)
Now, substitute these values in the expression.
2Step 2: Substitute and simplify
Replace the absolute values in the given expression with calculated values from step 1:
\(\sqrt{3}(2) + 3(\sqrt{3})\)
Now, we can simplify by using the distributive property:
\(2\sqrt{3} + 3\sqrt{3}\)
Combine the like terms (both terms have a common factor of \(\sqrt{3}\)):
\((2+3)\sqrt{3}\)
3Step 3: Evaluate the expression
Finally, add the numbers inside the parentheses and evaluate:
\(5\sqrt{3}\)
The given expression simplifies to \(5\sqrt{3}\).
Key Concepts
Evaluating ExpressionsDistributive PropertySimplifying Radicals
Evaluating Expressions
When it comes to evaluating expressions, the goal is to simplify the expression step by step until you reach a final answer. You evaluate expressions by substituting numbers for variables, performing operations, and simplifying as needed.
In the context of absolute value calculations, evaluating expressions involves a special consideration: the absolute value. It represents the distance of a number from zero on the number line, which is always a non-negative value. This means that when you encounter an absolute value in an expression, such as \( |-2| \), your first step is to determine its non-negative value, in this case, 2.
Once you’ve dealt with absolute values, you follow the usual order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). For the example \( \sqrt{3}|-2|+3|-\sqrt{3}| \), after evaluating the absolute values, you would proceed to multiply and then add the expressions according to these rules.
In the context of absolute value calculations, evaluating expressions involves a special consideration: the absolute value. It represents the distance of a number from zero on the number line, which is always a non-negative value. This means that when you encounter an absolute value in an expression, such as \( |-2| \), your first step is to determine its non-negative value, in this case, 2.
Once you’ve dealt with absolute values, you follow the usual order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). For the example \( \sqrt{3}|-2|+3|-\sqrt{3}| \), after evaluating the absolute values, you would proceed to multiply and then add the expressions according to these rules.
Distributive Property
The distributive property is a useful tool in algebra and simplifies expressions by distributing a single term across terms inside a parenthesis. It's represented as \( a(b + c) = ab + ac \).
Using this property can turn a complex-looking expression into simpler parts that are easier to manage. In our sample expression \( \sqrt{3}(2) + 3(\sqrt{3}) \), the distributive property isn't needed in its typical form since there are no parentheses enclosing a sum or difference. Instead, you recognize that each term includes a common factor of \( \sqrt{3} \) and combine them, effectively distributing the \( \sqrt{3} \) across the addition sign.
This principle can help you combine like terms effectively, turning \( 2\sqrt{3} + 3\sqrt{3} \) into \( (2+3)\sqrt{3} \), which further simplifies to \( 5\sqrt{3} \) when you evaluate the addition within the parentheses.
Using this property can turn a complex-looking expression into simpler parts that are easier to manage. In our sample expression \( \sqrt{3}(2) + 3(\sqrt{3}) \), the distributive property isn't needed in its typical form since there are no parentheses enclosing a sum or difference. Instead, you recognize that each term includes a common factor of \( \sqrt{3} \) and combine them, effectively distributing the \( \sqrt{3} \) across the addition sign.
This principle can help you combine like terms effectively, turning \( 2\sqrt{3} + 3\sqrt{3} \) into \( (2+3)\sqrt{3} \), which further simplifies to \( 5\sqrt{3} \) when you evaluate the addition within the parentheses.
Simplifying Radicals
Working with simplifying radicals can often come across as tricky for many students, but it's essentially about expressing the radical as simply as possible. A radical is an expression that includes a root, such as a square root \( \sqrt{} \) or a cube root \( \sqrt[3]{} \).
In the expression \( 5\sqrt{3} \) from our sample problem, the radical is already in its simplest form because \( \sqrt{3} \) cannot be simplified further - there are no perfect square factors in 3. In other cases, you may need to break down the number inside the radical into its prime factors and simplify from there. For example, \( \sqrt{16} \) simplifies to 4, because 16 is a perfect square (4x4).
It's important to remember that we can simplify radical expressions by combining like terms, just as we do with algebraic expressions, as long as the radicals have the same index and radicand (the number inside the root).
In the expression \( 5\sqrt{3} \) from our sample problem, the radical is already in its simplest form because \( \sqrt{3} \) cannot be simplified further - there are no perfect square factors in 3. In other cases, you may need to break down the number inside the radical into its prime factors and simplify from there. For example, \( \sqrt{16} \) simplifies to 4, because 16 is a perfect square (4x4).
It's important to remember that we can simplify radical expressions by combining like terms, just as we do with algebraic expressions, as long as the radicals have the same index and radicand (the number inside the root).
Other exercises in this chapter
Problem 32
Perform the indicated operations and simplify. $$ (5 m-2 n)(5 m+3 n) $$
View solution Problem 33
Perform the indicated operations and simplify. \(x-\frac{x^{2}}{x+2}+\frac{2}{x-2}\)
View solution Problem 33
Solve the equation by using the quadratic formula. $$ 4 x=-2 x^{2}+3 $$
View solution Problem 33
Carry out the indicated operation and write your answer using positive exponents only. $$ \left(\frac{x^{3}}{-27 x^{-6}}\right)^{-2 / 3} $$
View solution