Problem 92 - Introductory Algebra for College Students

Question

In Exercises \(89-92,\) express each sentence as a single numerical expression. Then use the order of operations to simplify the expression. Subtract 11 from \(9 .\) Multiply this difference by \(2 .\) Raise this product to the fourth power.

Step-by-Step Solution

Answer

The simplified expression is \(256\).

Step 1: Write down the mathematical expression

Firstly, convert the sentences into mathematical notation. 'Subtract 11 from 9' gives \(9 - 11\). 'Multiply this difference by 2' results in \((9 - 11) * 2\). 'Raise this product to the fourth power' gives \(((9 - 11) * 2) ^ 4\).

Step 2: Simplify the expression

Using the order of operations (BIDMAS / PEMDAS), we first calculate the operation in the brackets, which means subtracting 11 from 9 to get \(-2\). The expression then becomes \(-2 * 2 ^ 4\). Then, multiply \(-2\) by \(2\) to get \(-4\). This changes the expression to \(-4^4\). Finally, raise \(-4\) to the power of \(4\) to get the final result \(256\).

Key Concepts

Numerical Expression

A numerical expression is a mathematical phrase that includes numbers, operations, and sometimes variables, but does not have an equal sign. Unlike equations, numerical expressions do not express a relationship between two statements, but instead can be simplified down to a single numerical value.

For instance, in our example from the textbook, the sentences provided guide us to form the numerical expression \( (9 - 11) \times 2^4 \), which captures the operations to perform without presenting a comparative statement. The basis of working with numerical expressions is understanding the sequence of operations, which brings us to the crucial rule known as PEMDAS or BIDMAS in some regions.

Knowing how to correctly compose a numerical expression from a textual instruction is the first step towards mastering basic algebra and arithmetic skills. It often serves as the preliminary stage before delving into more complex equations and inequalities.

Simplifying Expressions

To simplify an expression means to perform all the possible operations to reduce it to its simplest form or a single number. This process often involves combining like terms, applying properties of operations, and executing operations in the correct sequence.

Take, for example, the expression \( ((9 - 11) \times 2)^4 \). Simplifying this expression requires adherence to the order of operations to ensure each step is completed in the right order for an accurate outcome.

When simplifying, it's important to avoid common mistakes, such as combining non-like terms or misapplying the operational rules. Simplification is a fundamental skill in algebra, which not only helps in solving mathematical problems but also in developing critical thinking and problem-solving strategies. It's the art of making complex expressions more approachable and understandable.

PEMDAS/BIDMAS

PEMDAS and BIDMAS are acronyms that represent the order of operations in mathematics: Parentheses/Brackets, Exponents/Indices, Multiplication, Division, Addition, and Subtraction. This order dictates the sequence in which operations should be performed in a numerical expression to ensure accuracy.

In our textbook example, we simplify the expression by following the PEMDAS/BIDMAS sequence. We first address the Parentheses/Brackets by performing the subtraction \(9 - 11\), yielding \-2\. Next, according to PEMDAS/BIDMAS, we handle the Exponents by raising \-2\ to the 4th power, after which we multiply. As a result, the expression simplifies to \(256\).

Understanding this hierarchy is crucial for students because it applies universally to mathematics. Misinterpretation of the order can lead to incorrect results. Thus, remember PEMDAS/BIDMAS, and you'll have a useful tool that structures the chaos of complex numerical expressions into manageable steps for simplification.