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Chapter 1: Problem 1

Evaluate each expression without using a calculator. \(\left(2^{2} \cdot 2\right)^{2}\)

Short Answer

Expert verified
The evaluated expression is 64.

Step by step solution

01

Simplify Inside the Parentheses

The inside of the parentheses has the expression \(2^2 \cdot 2\). First, calculate \(2^2\), which equals 4. Then, multiply 4 by 2 to get the result: \(4 \cdot 2 = 8\). Now, the expression inside the parentheses simplifies to \(8\).
02

Apply the Outer Exponent

Now we have the expression \(8^2\) after simplifying the inside. Calculate \(8^2\), which means multiplying 8 by itself: \(8 \times 8 = 64\). Hence, the result of the expression is \(64\).

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Key Concepts

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

Order of Operations
When solving mathematical expressions, order of operations is crucial. This concept helps to clarify which operations should be performed first to avoid confusion and ensure consistency in results. A common acronym used to remember this order is PEMDAS:
  • P: Parentheses 鈥� Always start solving expressions within the parentheses.
  • E: Exponents 鈥� Solve exponents (or powers) next.
  • MD: Multiplication and Division 鈥� From left to right.
  • AS: Addition and Subtraction 鈥� From left to right.

In the given exercise, we begin by simplifying the expression inside the parentheses \((2^{2} \cdot 2)\), respecting the first rule of order of operations. First we calculate the exponent \(2^2\), and then proceed to the multiplication within the parentheses.
Simplification
Simplification is the process of reducing an expression to its most basic form without changing its value. This makes complex problems easier to solve. In the exercise, the first step is to simplify inside the parentheses: \(2^{2} \cdot 2\). Begin by evaluating the exponent \(2^2 = 4\). Next, perform the multiplication: \(4 \cdot 2 = 8\).
Once the expression \(2^{2} \cdot 2\) is simplified to \(8\), we then simplify the outer expression \(8^2\). Reducing expressions like this not only helps to avoid mistakes but also allows you to see the structure and solution of the problem more clearly.
Arithmetic Operations
Arithmetic operations include basic functions such as addition, subtraction, multiplication, and division. In our context, multiplication plays a significant role in processing the expression.
In the exercise provided, you perform a series of arithmetic operations:
  • Exponentiation: Calculate powers like \(2^2 = 4\).
  • Multiplication: Multiply the resulting 4 by 2, which yields 8.
  • Exponentiation (again): Finally, compute \(8^2\) by multiplying 8 by itself to get 64.

Understanding how to properly perform and sequence these operations ensures accurate and efficient problem-solving in mathematics. When you break down each operation step by step, it becomes much easier to manage and solve even more complex mathematical problems.

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Most popular questions from this chapter

Graph the parabola \(y_{1}=1-x^{2}\) and the semicircle \(y_{2}=\sqrt{1-x^{2}}\) on the window \([-1,1]\) by \([0,1] .\) (You may want to adjust the window to make the semicircle look more like a semicircle.) Use TRACE to determine which is the "inside" curve (the parabola or the semicircle) and which is the "outside" curve. These graphs show that when you graph a parabola, you should draw the curve near the vertex to be slightly more "pointed" than a circular curve.

Write each interval in set notation and graph it on the real line. $$ (-3,5] $$

Write each interval in set notation and graph it on the real line. $$ (-\infty, 2] $$

Solve each equation by factoring. $$ 5 x^{4}=20 x^{3} $$

Write each interval in set notation and graph it on the real line. $$ [7, \infty) $$
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