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Chapter 5: Problem 11

Evaluate each expression. $$ 7 \cdot 2^{4} $$

Short Answer

Expert verified
The expression evaluates to 112.

Step by step solution

01

Evaluate the Exponent

First, we need to evaluate the expression inside the exponent. The expression is \(2^4\), which means 2 raised to the power of 4. Calculating \(2^4\):\(2^4 = 2 \times 2 \times 2 \times 2 = 16\)
02

Multiply by 7

Now, take the result from Step 1 and multiply it by 7 as indicated in the expression.Perform the multiplication:\(7 \times 16 = 112\)

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Key Concepts

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

Understanding Order of Operations
When dealing with mathematical expressions, it is crucial to follow the order of operations to arrive at the correct answer. This rule is often remembered by the acronym PEMDAS:
  • Parentheses - Complete any calculations inside parentheses first.
  • Exponents - Next, handle the exponents or powers.
  • Multiplication and Division - Perform these operations from left to right.
  • Addition and Subtraction - Finally, perform these operations, also from left to right.
For example, in the expression \[7 \cdot 2^{4}\], we need to evaluate the exponent, \(2^4\), before multiplying it by 7. This ensures that the operations are performed in the proper sequence and the expression is simplified correctly. Failure to follow this order can lead to incorrect results, especially as expressions become more complex and include a mix of different operations.
The Role of Multiplication
Multiplication is one of the fundamental operations in mathematics. It is essentially repeated addition, where a number is added to itself a specific number of times.
For example, in the expression \[7 \times 16\], the number 16 is essentially added 7 times.
  • It is denoted by different symbols such as "\( \times \)" or "\( \cdot \)".
  • It follows right after the evaluation of exponents in the order of operations.
To evaluate multiplication, you simply multiply the numbers as indicated. In the context of our expression, after calculating \(2^4 = 16\), you multiply the result by 7, getting\(7 \times 16 = 112\).Understanding multiplication in this way helps when it becomes part of a larger equation, ensuring each step is broken down and handled correctly.
Exploring Mathematical Expressions
Mathematical expressions are combinations of numbers, operators, and sometimes variables that represent a particular value or set of values. They are the language of mathematics and allow us to concisely convey complex concepts and operations.
In our exercise, the expression was \[7 \, \cdot \, 2^4\]. This includes:
  • A multiplication sign to indicate an operation that combines two quantities.
  • An exponent signifying repeated multiplication of a number, or a power.
Evaluating an expression requires following mathematical conventions such as the order of operations. Being familiar with these conventions ensures that you can read, interpret, and solve expressions effectively. It requires knowing how to break down each part, solve it step by step, and understand the relationships between the components involved in the expression.

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Most popular questions from this chapter

Subtract. $$ \left(0.7 x^{2}+0.2 x-0.8\right)-\left(0.9 x^{2}+1.4\right) $$

Subtract \(\left(4 y^{2}-6 y-3\right)\) from the sum of \(\left(8 y^{2}+7\right)\) and \((6 y+9)\)

Subtract. $$ \left(2 x^{2}+3 x-9\right)-(-4 x+7) $$

Add or subtract as indicated. $$ \left(4 x^{2}+y^{2}+3\right)-\left(x^{2}+y^{2}-2\right) $$

Simplify each expression by combining any like terms. $$ 3 x-5 x+7 $$
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