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Chapter 9: Problem 58

Simplify. $$ 10^{3}-\\{2 \times[5 \times 3-(4+2)]\\} $$

Short Answer

Expert verified
982

Step by step solution

01

Simplify inside the parentheses

Start with simplifying inside the parentheses. Calculate the expression inside the parentheses \(4+2\): \(4 + 2 = 6\)
02

Evaluate the expression inside the brackets

Next, simplify the expression inside the brackets, which now becomes \(5 \times 3 - 6\): \[5 \times 3 = 15\] \[15 - 6 = 9\]
03

Simplify the expression inside the curly braces

Now simplify the expression inside the curly braces \(2 \times 9\): \[2 \times 9 = 18\]
04

Subtract from the exponential value

Finally, subtract the result from the exponential value: \[10^{3} - 18\] \[1000 - 18 = 982\]

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Key Concepts

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

Order of Operations
To simplify mathematical expressions accurately, following the order of operations is crucial. This order ensures calculations are done correctly. The commonly used acronym **PEMDAS** helps remember this sequence:
  • **P**arentheses
  • **E**xponents
  • **M**ultiplication
  • **D**ivision
  • **A**ddition
  • **S**ubtraction
It is vital to perform operations inside parentheses first, followed by exponents, and then multiplication and division from left to right. Lastly, do addition and subtraction from left to right. In the exercise provided, this sequence was strictly followed to reach the correct result.
Exponents
Exponents are a way of expressing repeated multiplication of a number by itself. For example, in the expression \(10^{3}\), the '3' is the exponent, indicating that 10 is multiplied by itself 3 times. So, \(10^{3}=10 \times 10 \times 10 = 1000\). Understanding exponents is essential as it can simplify complex calculations notably. In our exercise, the exponent was the initial part of the expression evaluated, resulting in 1000 before any other operations.
Parentheses
Parentheses are used in mathematical expressions to indicate which operations should be performed first. Operations inside parentheses take precedence over others. Beyond simple parentheses (\text() ), expressions can also use brackets \text[] \text{} for nested grouping.
For instance, in the original exercise \(10^{3}-\{2 \times[5 \times 3-(4+2)]\}\), various levels of parentheses, brackets, and braces prioritize specific calculations. The innermost expression, \(4 + 2\), was simplified first. This intermediate result was then used in subsequent steps, demonstrating how parentheses direct the flow of computation to ensure each step is carried out in the correct order.

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