Problem 48 - Basic College Mathematics

Question

Simplify. $$ 5 \cdot(1+3)^{2}-10-2(5-4) $$

Step-by-Step Solution

Answer

68

Step 1: Simplify inside the parentheses

First, simplify any expressions inside parentheses. For this problem, we have two sets of parentheses: 1. \(1+3\) 2. \(5-4\). Simplifying these yields: 1. \(1+3=4\) 2. \(5-4=1\).

Step 2: Simplify exponents

Next, handle exponents. Substitute \(1+3\) with 4 to get \(4^{2}\): \( (1+3)^{2} = 4^{2} = 16 \).

Step 3: Perform multiplication

Then, perform any multiplications:1. \(5 \cdot\ 16 \) for the first term, which gives \(80\).2. \-2 \cdot 1\ for the last term, which gives \(-2\).

Step 4: Perform addition and subtraction

Finally, combine all the simplified terms together by performing addition and subtraction operations in order:\(80 - 10 - 2\).First, \(80-10 = 70\).Then, \(70-2 = 68\).

Key Concepts

order of operations

When simplifying mathematical expressions, it's crucial to follow the order of operations. This set of rules dictates the sequence in which operations should be performed to ensure consistency and accuracy.
The order of operations can be remembered using the acronym PEMDAS:

• P: Parentheses
• E: Exponents
• M/D: Multiplication and Division (from left to right)
• A/S: Addition and Subtraction (from left to right)

By adhering to this order, you can systematically break down complex problems into simpler steps. For example, in the given exercise: $$5 \times (1+3)^{2} - 10 - 2(5-4)$$, you start by simplifying inside the parentheses, then handle exponents, followed by multiplication, and finally, addition and subtraction.
This hierarchy ensures that the operations are executed in the correct sequence, leading to the accurate solution of such problems.

parentheses

Parentheses are used to group parts of an expression that should be evaluated first. They play a vital role in setting the correct order of operations.
In our exercise, there are two sets of parentheses: $$1 + 3$$ and $$5 - 4$$. The rule here is to simplify these expressions as the first step:

• Simplifying $$1 + 3$$ gives us 4
• Simplifying $$5 - 4$$ results in 1.

By calculating these inner expressions first, you ensure the following mathematical operations are correctly applied. Parentheses help clarify the structure of an expression, guiding you to break it down into manageable parts.
Always start simplifying expressions inside parentheses before moving on to other operations.

exponents

Once the parentheses are simplified, the next step is to deal with exponents. An exponent indicates how many times a number (the base) is multiplied by itself.
In our problem, after simplifying the parentheses, we have: $$ (1+3)^{2} \rightarrow 4^{2} $$, which means 4 multiplied by itself.
Thus, $$ 4^{2} = 16 $$.
Handling exponents early in the sequence is essential because they significantly impact the magnitude of the terms in your expression. Exponents transform the simpler base numbers into larger or smaller values, depending on the power involved. Always address them before proceeding with multiplication, division, addition, or subtraction.

mathematical simplification

Mathematical simplification involves reducing an expression to its simplest form. This process makes complex problems easier to understand and solve.
In our example, we follow the simplified steps through the order of operations:

• After handling the parentheses, we were left with: $$5 \times (4)^{2} - 10 - 2 \times (1)$$.
• Handling the exponents, we simplified this to: $$5 \times 16 - 10 - 2 \times 1$$.
• Next, performing the multiplication, we got: $$80 - 10 - 2$$.
• Finally, carrying out the addition and subtraction left us with the answer: $$80 -10 =70$$ then $$70 - 2 = 68$$.

Simplification helps in breaking down each component of the expression into understandable and solvable pieces. It's a methodical approach to ensure clarity and precision in mathematics.