Problem 52 - Introductory Algebra for College Students

Question

In Exercises \(29-72,\) use the order of operations to simplify each expression. $$\left[11-4\left(2-3^{3}\right)\right] \div 37$$

Step-by-Step Solution

Answer

The simplified result of the given expression is \(13.703\)

Step 1: Resolve the Exponents

In the given expression, start by solving the exponent within the parentheses, which is \(3^3\). This results in \(3^3 = 27\). So, the expression becomes: \(11 - 4(2 - 27) \div 37\)

Step 2: Solve Inside the Parentheses

Next, subtract \(27\) from \(2\) in the parenthesis. This gives \(-25\). So, the expression now becomes: \(11 - 4(-25) \div 37\)

Step 3: Multiply

Now, multiply \(-4\) by \(-25\) which equals \(100\). So, the expression now is: \(11 + 100 \div 37\)

Step 4: Perform the Division

Divide \(100\) by \(37\), this equals approximately \(2.703\). So, the expression now reads: \(11 + 2.703\)

Step 5: Perform the Addition

Lastly, add \(11\) and \(2.703\) together to get the final answer. The final result will be \(13.703\)

Key Concepts

Order of Operations

Understanding the order of operations is crucial when simplifying algebraic expressions. It's a fundamental principle that dictates the sequence in which mathematical operations should be performed to arrive at the correct answer. The widely accepted order is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right), often remembered by the acronym PEMDAS.

In the given exercise, to simplify \[11-4(2-3^{3}) \div 37\], we start by dealing with the expression within the parentheses, including the exponent. This takes priority over multiplication, division, addition, or subtraction that may also be present outside the parentheses. This structured approach prevents errors and ensures that everyone solves the expression consistently, arriving at the same, correct answer.

Exponents in Algebra

Exponents are a way to express repeated multiplication. For instance, the expression \(3^3\) signifies \(3\) multiplied by itself \(3\) times. In algebra, understanding how to work with exponents is key, especially within more complex expressions that combine various operations. When simplifying an algebraic expression with exponents, always resolve the exponents first before moving on to other operations.

Consider the exercise \[11-4(2-3^{3}) \div 37\], where we simplify \(3^3\) to \(27\) as step 1. If exponents were not correctly applied, one might mistakenly multiply \(3\) only once by \(3\), getting \(9\) instead of \(27\), which would lead to an incorrect result. Always handle the exponents right after any operations within parentheses for an accurate solution.

Arithmetic Operations

Multiplication and Division

After parentheses and exponents, multiplication and division are the next operations to be attended to. When simplifying an expression, if both operations are present, they are performed from left to right, as they appear in the expression.

In the example \(11-4(2-3^{3}) \div 37\), after processing the exponent and parentheses, we dealt with the multiplication \(4*-25\), then the division \(100 \div 37\).

Addition and Subtraction

The final step in the order of operations is to carry out any addition or subtraction, again proceeding from left to right. In our exercise, after dividing, we added the result to \(11\), yielding our final simplified expression. This adherence to the correct sequence is critical for ensuring the accurate execution of arithmetic operations.