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Chapter 4: Problem 70

Simplify. $$ 6 \times 0.9+0.1 \div 4-0.2^{3} $$

Short Answer

Expert verified
5.417

Step by step solution

01

- Calculate the multiplication

First, calculate the multiplication part of the expression: \[ 6 \times 0.9 = 5.4 \]
02

- Calculate the division

Next, compute the division: \[ 0.1 \div 4 = 0.025 \]
03

- Calculate the exponentiation

Now, compute the exponentiation: \[ 0.2^{3} = 0.008 \]
04

- Combine all computed values

Finally, add and subtract all the computed values together: \[ 5.4 + 0.025 - 0.008 \] Simplify: \[ 5.417 \]

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Key Concepts

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

Basic Arithmetic Operations
Basic arithmetic operations are the foundation of all mathematical computations. These operations include addition, subtraction, multiplication, and division.
Here are some key points:
  • Addition: Combining two or more numbers to get a sum.
  • Subtraction: Taking one number away from another to get a difference.
  • Multiplication: Repeated addition of the same number.
  • Division: Splitting a number into equal parts.
In our exercise, we see all these operations:
  • The multiplication of 6 and 0.9, yielding 5.4.
  • The division of 0.1 by 4, resulting in 0.025.
  • The addition and subtraction of these results, giving a final simplified value of 5.417.
Order of Operations
The order of operations is crucial in simplifying expressions accurately. This set of rules dictates the correct sequence to evaluate a mathematical expression to ensure everyone gets the same result.
It is commonly remembered by the acronym PEMDAS:
  • Parentheses
  • Exponents (including square roots, cube roots, etc.)
  • Multiplication and Division (left to right)
  • Addition and Subtraction (left to right)
In our exercise:
  • First, we tackled multiplication: \text{6 \(\times\) 0.9}
  • Next, we performed division: \text{0.1 \(\times\) 4}
  • Then, exponentiation: 0.2鲁
  • Finally, we handled addition and subtraction: 5.4 + 0.025 - 0.008
Exponentiation
Exponentiation refers to the process of raising a number to the power of another number. For example, in the expression 0.2鲁, 0.2 is the base and 3 is the exponent. This means:
  • 0.2 multiplied by itself three times:
  • 0.2 \(\times\) 0.2 \(\times\) 0.2 = 0.008
Exponentiation follows specific rules:
  • Any number raised to the power of 1 remains the same.
  • Any number raised to the power of 0 equals 1.
Understanding these rules helps in accurately simplifying expressions involving powers. In our expression, we saw 0.2鲁 yielding 0.008.
Combining this with our other arithmetic results:
鈬� 5.4 + 0.025 - 0.008, gives us the simplified value of 5.417.

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