Problem 22 - Algebra 1

Question

Evaluate the exponential expression. Write fractions in simplest form $$8^{3} \cdot 0^{-1}$$

Step-by-Step Solution

Answer

The given expression is undefined due to the division by zero in the term \(0^{-1}\)

Step 1: Calculate \(8^{3}\)

Calculate \(8^{3}\) by multiplying 8 with itself thrice, which equals to 512.

Step 2: Interpret \(0^{-1}\)

The term \(0^{-1}\) means 1 divided by 0. But, division by zero is undefined in mathematics.

Step 3: Multiplication

No matter what value we got in step 1, if we multiply it by something undefined (like in step 2), the whole expression is undefined.

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication compactly. When you see a number with a small raised number beside it, that small number is the exponent. In the expression \(8^3\), 8 is the base, and 3 is the exponent. This means you multiply 8 by itself a total of three times: \(8 \times 8 \times 8 = 512\).

This helps to simplify writing very large or very small numbers but requires understanding how to multiply repeatedly.
When working with exponents:

• Remember that any number raised to the power of 1 is itself. For example, \(8^1 = 8\).
• Another key rule is that any number raised to the power of 0 is 1, this is known as the zero exponent rule.

It's also important to practice with different bases and exponents to become comfortable with these calculations.

The Challenge of Division by Zero

Division by zero presents a unique situation in mathematics. When you try to divide any number by zero, like in the expression \(0^{-1}\), it implies "1 divided by 0". This is not possible, as it doesn’t yield a result that can be defined in mathematical terms.

Think of division as the process of determining how many times one number is contained within another. But if you try to divide by zero, the question becomes, "how many zeros can fit into 1?"

• This leads to undefined and non-sensical outcomes.
• In calculations, if you end up with a division by zero, the entire operation cannot be completed and is marked as undefined.

Recognizing division by zero is crucial in avoiding incorrect solutions and understanding where mistakes may arise.

Combining Mathematical Operations

Mathematical expressions often involve multiple operations, like exponentiation, multiplication, and division. It's important to respect the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order tells us how to tackle expressions step by step.

In the given exercise, the expression is \(8^3 \cdot 0^{-1}\). According to the order, we first solve the exponent to get 512. Then, we approach the multiplication with the undefined division result from \(0^{-1}\).

• If any part of the expression results in an undefined value, the whole expression is affected.
• Operations like multiplication default to undefined if one factor is undefined, no matter how precise the other factors are.

This example illustrates the importance of careful step-by-step analysis in mathematical problem-solving.