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Chapter 5: Problem 13

Evaluate each expression. See Example 1. $$ 7 \cdot 2^{5} $$

Short Answer

Expert verified
224

Step by step solution

01

Evaluate the Exponent

First, solve for the exponent in the expression. The expression given is \(2^{5}\). Calculate \(2^5\) by multiplying 2 by itself four more times: \[ 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \]
02

Multiply by the Coefficient

Next, use the result from Step 1 to multiply it by the coefficient in the expression. The expression given is \(7 \cdot 32\). Calculate by performing the multiplication:\[7 \cdot 32 = 224\]
03

Conclusion

The evaluated result of the expression \(7 \cdot 2^5\) is 224.

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Key Concepts

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

Order of Operations
The order of operations is a fundamental principle in mathematics that determines the sequence in which operations should be performed. It ensures that everyone gets the same result when evaluating an expression. The commonly used acronym to remember this is PEMDAS:
  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
In the given expression, \(7 \cdot 2^5\), the order of operations tells us to evaluate the exponent first. This is because exponents come before multiplication in PEMDAS. Following this order avoids mistakes and ensures consistent results.
Multiplication
Multiplication is a mathematical operation where we add a number to itself a certain number of times. For example, if we multiply 7 by 32, we are essentially adding 7 together 32 times. This operation is symbolized by the "\( \cdot \)" or "脳" sign.In the expression \(7 \cdot 2^5\), once we have solved for the exponent, we then multiply the base value (32, which is the result of \(2^5\)) by 7. The result of \(7 \times 32\) is calculated as follows:\[ 7 \times 32 = 224 \]This step illustrates how multiplication is an efficient process of repeated addition, transforming complex problems into simpler steps.
Evaluating Expressions
Evaluating expressions involves finding the value of a mathematical phrase or equation by performing arithmetic operations in the correct sequence. It requires a clear understanding of the order of operations, as well as accurate calculation skills.In the exercise provided, evaluating the expression \(7 \cdot 2^5\) involves several steps:
  • First Step: Calculate the exponent. Solve \(2^{5}\), which equals 32.
  • Second Step: Multiply the result by 7, resulting in \(7 \cdot 32 = 224\).
By carefully applying each operation in sequence, we ensure that the expression is evaluated correctly, ultimately arriving at the numerical value of 224.

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Most popular questions from this chapter

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