Problem 27
Question
In the following exercises, simplify each expression. $$ 2[1+3(10-2)] $$
Step-by-Step Solution
Verified Answer
The simplified expression is 50.
1Step 1 - Simplify inside the parentheses
Start by simplifying the expression inside the innermost parentheses: \(10 - 2 = 8\) Now the expression becomes \[2[1 + 3(8)]\].
2Step 2 - Simplify inside the brackets
Next, multiply the numbers inside the brackets: \(3 \times 8 = 24\) Now the expression becomes \[2[1 + 24]\].
3Step 3 - Add inside the brackets
Next, add the numbers inside the brackets: \(1 + 24 = 25\) Now the expression becomes \[2[25]\].
4Step 4 - Multiply outside the brackets
Finally, multiply the number outside the brackets with the simplified result inside: \(2 \times 25 = 50\).
Key Concepts
Order of OperationsDistributive PropertyParentheses
Order of Operations
The 'order of operations' is crucial when simplifying algebraic expressions. It helps us determine the correct sequence in which to perform arithmetic operations. The standard rule to remember is PEMDAS:
In our example, the order of operations guides us to first simplify the innermost parentheses \(10 - 2\), then multiply \(3 \times 8\), add \(1 + 24\), and finally multiply \(2 \times 25\). Without this proper sequence, the result would be incorrect.
- P: Parentheses
- E: Exponents
- M/D: Multiplication and Division (from left to right)
- A/S: Addition and Subtraction (from left to right)
In our example, the order of operations guides us to first simplify the innermost parentheses \(10 - 2\), then multiply \(3 \times 8\), add \(1 + 24\), and finally multiply \(2 \times 25\). Without this proper sequence, the result would be incorrect.
Distributive Property
The distributive property links multiplication and addition/subtraction in algebra. It states that multiplying a number by a sum (or difference) is the same as doing each multiplication separately and then adding (or subtracting) the results. Mathematically, it is expressed as:
\[a(b + c) = ab + ac\]
In our solved problem, the distributive property is indirectly applied when we handle the multiplication inside the brackets:
\[a(b + c) = ab + ac\]
In our solved problem, the distributive property is indirectly applied when we handle the multiplication inside the brackets:
- We first simplify \(3(10 - 2)\), giving us the term \(24\).
- Next, by adding \(1\) to \(24\), we get \(25\).
- Finally, multiplying the outside term, \(2\), gives us the end result \(50\).
Parentheses
Parentheses are symbols used in algebra to group parts of an expression. This grouping indicates that the operations within the parentheses should be performed first, following the order of operations.
In our exercise, the innermost parentheses \(10 - 2\) are simplified first, highlighting the importance of dealing with nested expressions.
By solving \(10 - 2 = 8\), we re-write the expression as:
\[2[1 + 3(8)]\]
This process continues by further simplifying step-by-step, ensuring each part of the grouped expression is correctly handled. Remember, whenever you see parentheses, you start there as guided by the order of operations.
In our exercise, the innermost parentheses \(10 - 2\) are simplified first, highlighting the importance of dealing with nested expressions.
By solving \(10 - 2 = 8\), we re-write the expression as:
\[2[1 + 3(8)]\]
This process continues by further simplifying step-by-step, ensuring each part of the grouped expression is correctly handled. Remember, whenever you see parentheses, you start there as guided by the order of operations.
Other exercises in this chapter
Problem 25
In the following exercises, simplify each expression. $$ 3(1+9 \cdot 6)-4^{2} $$
View solution Problem 26
In the following exercises, simplify each expression. $$ 5(2+8 \cdot 4)-7^{2} $$
View solution Problem 28
In the following exercises, simplify each expression. $$ 5[2+4(3-2)] $$
View solution Problem 29
In the following exercises, simplify each expression. $$ 8+2[7-2(5-3)]-3^{2} $$
View solution