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Chapter 1: Problem 6

Simplify. Show your steps. $$7^{2}+5 \cdot 6 \div 10$$

Short Answer

Expert verified
Question: Simplify the expression: $$7^{2}+5 \cdot 6 \div 10$$ Answer: $$52$$

Step by step solution

01

Evaluate the Exponent

We start by evaluating the power: $$7^2 = 7 \cdot 7 = 49$$ The expression now becomes: $$49 + 5 \cdot 6 \div 10$$
02

Perform Multiplication and Division

Now we perform the multiplication and division operations from left to right: $$5 \cdot 6 = 30$$ $$30 \div 10 = 3$$ The expression now becomes: $$49 + 3$$
03

Perform Addition

Finally, we perform the addition operation: $$49 + 3 = 52$$ So, the simplified expression is: $$52$$

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Key Concepts

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

Order of Operations
When working with mathematical expressions, the order in which you perform calculations is crucial. This is where the order of operations comes into play. To avoid ambiguity and ensure consistency, mathematicians have agreed on a particular sequence called PEMDAS:
  • Parentheses first
  • Exponents (i.e., powers and square roots, etc.)
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
In our exercise, \(7^{2}+5 \cdot 6 \div 10\), we begin by addressing the exponents, followed by multiplication and division, and end with addition, as dictated by the PEMDAS rule. Remembering PEMDAS can help you simplify complex expressions correctly.
Exponents
Exponents are a shorthand way to express repeated multiplication of the same number. The number \(7^{2}\) is read as 'seven squared' and signifies \(7 \times 7\). Here's a quick breakdown of how exponents work:
  • \[a^n\] means multiplying \( a \) by itself \(n\) times.
  • The number \(a\) is the base, and \(n\) is the exponent.
  • Exponents are not limited to squares; they can be any integer鈥攑ositive, negative, or zero.
When simplifying expressions, it's important to evaluate exponents early on (after dealing with any parentheses) because they influence the magnitude of the numbers you'll be working with in subsequent steps.
Arithmetic Operations
The basic arithmetic operations include addition, subtraction, multiplication, and division. Each operation has a different level of precedence, as mentioned above with the order of operations. Here's a deeper insight into these operations:
  • Multiplication and division are on the same level of precedence鈥攚hichever comes first from left to right is addressed first.
  • Addition and subtraction are also of equal precedence and are addressed after multiplication and division, again from left to right.
  • In our exercise, once the exponent is evaluated, we tackle the multiplication \(5 \times 6\) and division \(30 \div 10\) before we perform the last step of addition to arrive at the simplified result of 52.
By understanding each operation and its precedence, you can systematically approach and simplify any mathematical expression.

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