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Chapter 9: Problem 60

Simplify. $$ 4^{1} \cdot\left(2^{3}-8\right) $$

Short Answer

Expert verified
The simplified result is 0.

Step by step solution

01

Evaluate the Exponentiation

First, simplify the exponentiation within the expression. The base 4 raised to the power of 1 is simply the number itself. Thus, \(4^1 = 4\).
02

Simplify the Expression in Parentheses

Next, simplify the expression inside the parentheses. Start by evaluating the exponentiation: \(2^3\) means \(2 \times 2 \times 2 = 8\). So, the expression becomes \(8 - 8\).
03

Perform the Subtraction

Subtract the numbers inside the parentheses: \(8 - 8 = 0\).
04

Final Multiplication

Multiply the simplified exponentiated number by the result from the parentheses: \(4 \times 0 = 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Exponentiation
Exponentiation is a fundamental concept in mathematics, the process of multiplying a number by itself a specified number of times. When we see something like \(4^1\), it indicates that the base number 4 is multiplied by itself 1 time. This simplifies to 4, as multiplying any number once results in the number itself.
Exponentiation becomes a bit more complex as the exponent increases. For instance, \(2^3\) means the number 2 is used as a factor three times. Instead of adding repeatedly, we use multiplication: \(2 \times 2 \times 2\), which equals 8. This demonstrates how exponentiation can compress long multiplication into a concise form.
Mastering the Order of Operations
When simplifying a mathematical expression, it is crucial to follow the order of operations to achieve the correct result. The acronym PEMDAS can help you remember this order:
  • Parentheses
  • Exponents
  • Multiplication
  • Division
  • Addition
  • Subtraction
The order of operations ensures everyone solves a mathematical problem the same way. This concept applies directly to our example: we begin by resolving anything in parentheses, then calculate exponents, and finally tackle multiplication and addition/subtraction in the order they appear from left to right.
In the exercise given, we first dealt with the exponent outside the parentheses, then proceeded to the subtraction inside the parentheses, ensuring we didn't jumble steps.
Importance of Parentheses in Mathematics
Parentheses are used in mathematics to specify which operations should be performed first within an expression. They act like a guidepost for your calculations, directing the flow and influencing the ultimate result. By clustering parts of an equation together, operations within parentheses override the "normal" order defined by PEMDAS.
In our problem, the expression within parentheses \((2^3 - 8)\) needed resolving before carrying out any further operations. We focused first on exponentiation within the parentheses, simplifying \(2^3\) into 8, before tackling the subtraction, making the parentheses a crucial aspect of getting to the correct answer.

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