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Chapter 11: Problem 53

VDetermine if each is a valid arithmetic expression. $$-(3 * 4 \uparrow 5)$$

Short Answer

Expert verified
The given expression is $$-(3 * 4 \uparrow 5)$$. Following the order of operations, first apply exponentiation: \(4 \uparrow 5 = 4^5 = 1024\). Next, multiplication: \(3 * 1024 = 3072\). Finally, the subtraction indicated by the negative sign: \(-3072\). Since we can perform all arithmetic operations without issues, the given expression is a valid arithmetic expression.

Step by step solution

01

1. Interpret Arithmetic Operations and Symbols

In the given expression, $$-(3 * 4 \uparrow 5),$$ there are three arithmetic operations involved: 1. Subtraction: Represented by the negative sign " - ". 2. Multiplication: Represented by the asterisk " * ". 3. Exponentiation: Represented by the arrow " \( \uparrow \) ".
02

2. Analyze the Order of Operations

According to the order of operations (PEMDAS/BODMAS), the operations should be carried out in the following order: 1. Parentheses (Brackets) 2. Exponents (Orders, Roots) 3. Multiplication and Division (from left to right) 4. Addition and Subtraction (from left to right)
03

3. Apply the Order of Operations to the Given Expression

Now, let's apply the order of operations to the given expression: $$-(3 * 4 \uparrow 5)$$ First, let's deal with the exponentiation operation: $$4 \uparrow 5 = 4^5$$ Now, apply the exponentiation: $$4^5 = 1024$$ Next, apply the multiplication operation: $$3 * 1024 = 3072$$ Finally, apply the subtraction operation, which is denoted by the negative sign " - " to the result: $$-3072$$ Therefore, the given expression is a valid arithmetic expression, as we were able to perform the arithmetic operations in the correct order without encountering any issues.

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Key Concepts

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

Order of Operations
Understanding the order of operations is crucial for solving arithmetic expressions correctly. It's a set of rules that specify which procedures to perform first in a mathematical expression. Commonly remembered by the acronym PEMDAS or BODMAS, it outlines the sequence to follow: Parentheses first, then Exponents (including roots, represented by up arrows in some expressions), followed by Multiplication and Division (from left to right), and lastly, Addition and Subtraction (from left to right).

When looking at our exercise, \( -(3 * 4 \uparrow 5) \), it's essential to identify each operation and its order. Despite the expression starting with a subtraction symbol, it doesn't mean we subtract first. We follow the order of operations, starting with exponentiation, multiplication, and then addressing the negative sign, which is part of the subtraction family. By adhering to these rules, we avoid common mistakes and ensure the expression is evaluated correctly.
Exponentiation
Exponentiation is an arithmetic operation where a number, known as the base, is multiplied by itself a certain number of times indicated by the exponent. The expression \( 4 \uparrow 5 \) represents 4 raised to the power of 5, or \( 4^5 \). This means 4 multiplied by itself 4 more times. It is important to distinguish exponentiation from multiplication or addition as its effect on numbers is much more substantial.

In practice, to solve \( 4 \uparrow 5 \), we multiply 4 by itself five times: \( 4 \times 4 \times 4 \times 4 \times 4 \), which equals 1024. This step is fundamental before moving on to any other operations, as dictated by the order of operations. Exponentiation can dramatically change the size of a number, and it鈥檚 essential for students to grasp its concept to succeed in mathematics.
Arithmetic Operations
Arithmetic operations are the building blocks of mathematics. They include addition, subtraction, multiplication, division, and exponentiation. These operations help us solve a wide range of problems, from simple calculations to complex equations.

Each operation has a specific symbol to denote it:
  • Addition is symbolized by \( + \).
  • Subtraction is symbolized by \( - \).
  • Multiplication by \( \times \) or an asterisk \( * \).
  • Division by \( \div \) or a slash \( / \).
  • Exponentiation, as seen in the exercise, can be symbolized by an up arrow \( \uparrow \), though it is more commonly represented by a raised number.

In our example expression \( -(3 * 4 \uparrow 5) \), we integrate several arithmetic operations, each with a clear hierarchy defined by the order of operations. Understanding each operation and its notation ensures that students can accurately solve arithmetic expressions without confusion.

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Most popular questions from this chapter

Use the grammar \(G=(N, T, P, \sigma),\) where \(N=\\{A, \sigma\\}, T=\\{a, b\\},\) and \(P=\\{\sigma \rightarrow a \sigma, \sigma \rightarrow a A, A \rightarrow b\\},\) to answer Exercises \(15-23\) . Draw a derivation tree for each word in \(L(G)\) . $$\mathrm{a}^{4} \mathrm{b}$$

Find three words belonging to each language over \(\sigma=\\{0,1\\}\). \(A \Lambda=\Lambda A\)

Develop a grammar that generates each language over {0,1}. $$\\{1,11,1111,11111111, \ldots\\}$$

Use the grammar \(G=(N, T, P, \sigma),\) where \(N=\\{A, \sigma\\}, T=\\{a, b\\},\) and \(P=\\{\sigma \rightarrow a \sigma, \sigma \rightarrow a A, A \rightarrow b\\},\) to answer Exercises \(15-23\) . Draw a derivation tree for each word in \(L(G)\) . $$\mathrm{a}^{3} \mathrm{b}$$

For Exercises \(68-73,\) use the following definition of a simple algebraic expression: $$\langle\text {expression}\rangle : :=\langle\text { term }\rangle |\langle\text { sign }\rangle\langle\text { term }\rangle |$$ $$\langle\text { expression }\rangle\langle\text { adding operator }\rangle\langle\text { term }\rangle$$ $$\langle\operatorname{sign}\rangle \therefore=+ 1-$$ $$\langle\text { adding operator}\rangle: :=+1-$$ $$\langle\text { term }\rangle : :=\langle\text { factor }\rangle |$$ $$\langle\text { term }\rangle\langle\text { multiplying operator }\rangle\langle\text { factor }\rangle$$ $$\langle\text { multiplying operator }\rangle := *| /$$ $$\langle\text { factor }\rangle : :=\langle\text { letter }|\rangle (\langle\text { expression }\rangle |\langle\text { expression }\rangle$$ $$\langle\text { letter }\rangle : := a|b| c | \ldots : z$$ Determine if each is a legal expression. $$a+b *(c / d)$$
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