Problem 53 - Elementary Algebra

Question

Use the order of operations to simplify the quantities for the following problems. $$ 2^{5}+3(8+1) $$

Step-by-Step Solution

Answer

Answer: The simplified result of the expression is 59.

Step 1: Identify the Parts of the Expression that Involve Parentheses

In the expression, we have one set of parentheses: \((8+1)\). We will simplify the expression within the parentheses first.

Step 2: Simplify the Expression within the Parentheses

Inside the parentheses, we have the addition operation: \((8+1)\). We simplify it to obtain: \((9)\).

Step 3: Simplify the Exponent

In the expression, we have an exponent \(2^5\). Since exponents come next after parentheses in the order of operations, we simplify the exponent by calculating \(2^5 = 32\).

Step 4: Perform Multiplication

Next, we deal with the multiplication operation: \(3(9)\), which comes from an earlier step where we simplified the expression within the parentheses. We multiply \(3\) by \(9\) to obtain \(3(9) = 27\).

Step 5: Perform Addition

Finally, we add the results of the previous steps. We have \(32+27\), which yields \(32+27 = 59\). Thus, the simplified result for the given expression is \(59\).

Key Concepts

Simplifying Expressions

Simplifying expressions in mathematics is akin to cleaning up and organizing a cluttered room, making it easier to understand what's in it. The goal is to condense the expression into its simplest form using basic operations such as addition, subtraction, multiplication, division, as well as exponents and parentheses.

Imagine you have a math phrase, like a sentence, and simplifying is how you make that sentence shorter and clearer without changing its meaning. For students, mastering this concept is crucial because it not only makes the individual parts of a problem more manageable but also prepares you for more advanced topics such as algebra and calculus.

Here's a tip: always work step by step, focusing on one operation at a time, and make sure to refer back to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This strategy helps prevent errors and ensures that each part of the expression gets the attention it deserves.

Exponents

Exponents are like shorthand for repeated multiplication. They tell you how many times to use a number in a multiplication. In the expression 2^5, 2 is the base, and 5 is the exponent.

Understanding how to handle exponents is essential because they're not just about making numbers larger. They can represent growth, area, volume, and more in real-world situations, like calculating interest in a bank account or determining the number of cells in a growing organism.

To simplify exponents as in 2^5, multiply 2 by itself 5 times, which is 2*2*2*2*2 resulting in 32. Remember, an exponent applied to a parenthesis affects everything inside it, so (2+3)^2 is not 2^2 + 3^2, but (5)^2, or 25. Always perform exponentiation before you multiply, divide, add, or subtract, unless the order is changed by parentheses!

Parentheses in Math

Parentheses in math are like signals on a road—they guide you on what to tackle first. They're used to group numbers and operations, indicating that operations within them should be performed before anything outside. This can drastically change the outcome of an expression. For instance, 3(8+1) means add 8+1 first, then multiply by 3, giving 27.

It's crucial to follow the 'innermost first' rule. If you have nested parentheses, like (3(2+1)), start with the inner pair: 2+1. Once simplified, you move outward. In that case, you would first get (3*3), then simplify further to 9.

Parentheses can also be used to override the conventional order of operations. For example, in 2 + (3*4), you would multiply before adding, despite the usual rule of performing multiplication after addition.