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Chapter 4: Problem 88

Simplify. $$ 4^{2} \cdot 5 $$

Short Answer

Expert verified
The simplification of \(4^2 \cdot 5\) is 80.

Step by step solution

01

Calculate the Exponent

First, solve the exponent part of the expression. Calculate \(4^2\), which means multiply 4 by itself: \(4 \times 4 = 16\).
02

Multiply the Results

Now multiply the result from Step 1 with the number 5. Calculate \(16 \times 5\): \(16 \times 5 = 80\).

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Key Concepts

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

Simplifying Expressions
Simplifying mathematical expressions is about reducing them to their simplest form without changing their value. Doing this helps make complex problems easier to solve. In any expression involving multiplication and exponentiation, follow these guidelines to simplify effectively:
  • Always respect the operation hierarchy, often referred to as the order of operations (PEMDAS/BODMAS).
  • Handle any operations involving exponents before moving on to multiplication.
  • Aim to convert complex expressions into simple, understandable computations.
For instance, in the expression \(4^2 \cdot 5\), you address the exponent \(4^2\) first to get 16, simplifying the multiplication process that follows.
Multiplication
Multiplication is a fundamental arithmetic operation that combines equal groups. It simplifies the process of repeated addition. In our example, after simplifying the exponent, we proceed with multiplication:
  • The expression becomes \(16 \times 5\).
  • Understand this as adding 16 together 5 times or adding 5 together 16 times, which ever makes it easier.
  • Compute step-by-step: \(16 + 16 = 32\), \(32 + 16 = 48\), \(48 + 16 = 64\), \(64 + 16 = 80\).
Thus, 16 multiplied by 5 yields 80. Breaking multiplication into simpler steps can help in understanding how products result from addition.
Order of Operations
The order of operations is a set of rules that dictates the sequence in which calculations should be performed in a mathematical expression. This sequence is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) or BODMAS in some regions (Brackets, Orders, Division, Multiplication, Addition, Subtraction). These rules ensure consistency in solving expressions.

In our example, this sequence guides us:
  • Exponents: Start by solving any exponential expressions, such as \(4^2\), giving us 16.
  • Multiplication: Next, carry out multiplication, \(16 \times 5\), to find the final result of 80.
By following this order, you ensure expressions are simplified correctly, leading to accurate results. Skipping steps or altering the sequence may lead to errors in computation.

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Most popular questions from this chapter

Simplify. See Section 1.7. $$5^{3}$$

Recall that to find the average of two numbers, we find their sum and divide by \(2 .\) For example, the average of \(\frac{1}{2}\) and \(\frac{3}{4}\) is \(\frac{\frac{1}{2}+\frac{3}{4}}{2}\). Find the average of each pair of numbers. Study Exercise \(67 .\) Without calculating, \(\operatorname{can} \frac{1}{3}\) be the average of \(\frac{1}{2}\) and \(\frac{8}{9}\) ? Explain why or why not.

Write each improper fraction as a mixed number or a whole number. See Example 21. $$\frac{64}{9}$$

Solve. See the Concept Checks in this section. Round the mixed number \(5 \frac{11}{12}\) to the nearest whole number

Write each mixed number as an improper fraction. See Example 20. $$9 \frac{7}{20}$$
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