Problem 53 - Algebra 1

Question

Simplify the variable expression. $$-(2)^{3}(b)$$

Step-by-Step Solution

Answer

The simplified form of the variable expression \(-(2)^{3}(b)\) is \(-8b\).

Step 1: Evaluate the Power.

Begin by calculating the power, i.e., \(-(2)^{3}\) = \(-8\). This is because, in the order of operations, we do the power calculation before multiplication.

Step 2: Perform the Multiplication.

The next stage is to execute the multiplication, i.e., \(-8 * b\), which is simply \(-8b\).

Step 3: Final Simplification.

The final simplified form of the given expression \(-(2)^{3}(b)\) is \(-8b\).

Key Concepts

Understanding the Order of Operations

The order of operations is a fundamental concept in mathematics that dictates the sequence in which mathematical operations should be performed to correctly simplify expressions. It's often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

When simplifying expressions, you must first evaluate any calculations inside parentheses. Next, you handle exponents, which represent repeated multiplication. Only after these two steps do you proceed to multiplication and division, which are performed from left to right as they appear in the expression. Lastly, you tackle addition and subtraction in the order they appear. Neglecting this structured sequence could lead to incorrect results.

For example, in the expression \( -(2)^{3}(b) \), we start by evaluating the exponentiation before moving on to the multiplication by the variable \( b \). It's crucial for students to approach simplifying expressions by strictly adhering to these rules of the order of operations, as showcased in the step-by-step solution.

Exponentiation and Its Rules

Exponentiation is a form of mathematical shorthand for expressing a number multiplied by itself a certain number of times. The number being multiplied is called the 'base,' and the number of times it is multiplied by itself is the 'exponent.' When you see an expression like \( 2^3 \), it means 2 is the base and 3 is the exponent, indicating that 2 is to be multiplied by itself 3 times, resulting in \( 2 \times 2 \times 2 \), which equals 8.

Understanding exponentiation is crucial because it's a common operation that can greatly affect the value of an algebraic expression. A negative base raised to an odd exponent will result in a negative result, as in the example of \( -(2)^3 \), which simplifies to -8. Conversely, a negative base raised to an even exponent results in a positive product. This distinction is key when simplifying variable expressions involving exponents.

Working with Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \(b\) in our example), and arithmetic operations. In addition to understanding exponentiation and order of operations, recognizing how to handle the variables is essential in simplifying algebraic expressions.

Variables represent unknown values and can be manipulated just like numbers in an expression. The key is to perform the same operations on the variables as you would with numbers, while also keeping in mind that unlike numbers, variables cannot be combined unless they represent the same quantity.

When it's time to multiply a number with a variable, such as \( -8 \times b \), simply attach the variable to the number, resulting in \( -8b \). This multiplication does not alter the variable, but scales it by the number. Ensuring clarity and comfort in these basic manipulations with variables will help students to tackle more complex algebraic expressions with confidence.