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Magazine Chapter 2: Problem 63
For Exercises \(1-74,\) simplify. $$\\{|-12(-5)-38|+2\\} \div 3[-9+\sqrt{49}]^{3}$$
Short Answer
Expert verified
Question: Simplify the expression |-12(-5)-38|[{-9+sqrt{49}}]^3 梅 3.
Answer: $$-\frac{176}{3}.$$
Step by step solution
01
Evaluate the expression inside the absolute value
Begin by evaluating the expression inside the absolute value: |-12(-5)-38|.
To do this, first multiply -12 by -5, and then subtract 38 from the result.
|-12(-5)-38| = |60 - 38|
02
Calculate the absolute value
Now, calculate the absolute value of the expression inside the brackets: |60 - 38| = |22|.
The absolute value of a number is its distance from zero, so |22| = 22.
03
Evaluate the expression in square brackets
Next, we need to evaluate the expression inside the square brackets: [-9 + sqrt{49}].
To do this, first find the square root of 49, and then add it to -9.
-9 + sqrt{49} = -9 + 7
04
Simplify the expression in square brackets
Now, simplify the expression inside the square brackets: -9 + 7 = -2.
05
Calculate the expression with the exponent
Now, we need to raise the result from step 4 to the power of 3: (-2)^3.
To do this, multiply -2 by itself three times:
(-2)^3 = -2 脳 -2 脳 -2 = -8
06
Divide the absolute value by 3
Now, divide the absolute value result (22) by 3:
{22} 梅 3 = \frac{22}{3}
07
Complete the expression
Finally, multiply the result from step 6 by the result from step 5:
(\frac{22}{3}) 脳 (-8) = -\frac{176}{3}
The simplified expression is: $$-\frac{176}{3}$$.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Absolute Value
The absolute value of a number indicates its distance from zero on a number line. It is always a non-negative value, regardless of whether the number itself is negative or positive.
For example, the absolute value of \(-5\) is \(|-5| = 5\). This is because \(-5\) is 5 units away from zero.
For example, the absolute value of \(-5\) is \(|-5| = 5\). This is because \(-5\) is 5 units away from zero.
- The absolute value can be represented by vertical bars, such as \(|x|\).
- When simplifying arithmetic expressions, absolute values are handled as a positive transformation of the number inside.
- In the original exercise, the expression inside the absolute value is \(-12(-5)-38\),\ which simplifies to \(|60 - 38| = |22|\).
Square Root
The square root of a number \( n \) is the value that, when multiplied by itself, produces \( n \). Understanding square roots is essential when solving arithmetic expressions, especially when dealing with radical expressions like \( \sqrt{49} \), which equals \(7\).
Here's how you calculate:
Here's how you calculate:
- Identify the number for which you need the square root.
- Break it down to find a number that, when squared, gives the original number.
- The square root of a perfect square is an integer, e.g., \(\sqrt{49} = 7\).
Exponents
Exponents are used to represent repeated multiplication of a number by itself. The expression \(a^n\) denotes the base \(a\) raised to the power of \(n\), meaning \(a\) is multiplied by itself \(n\) times.
For example, the expression \((-2)^3\) means \(-2\) is multiplied by itself three times, resulting in:
For example, the expression \((-2)^3\) means \(-2\) is multiplied by itself three times, resulting in:
- \((-2)\times (-2) = 4\)
- \(4 \times (-2) = -8\)
Order of Operations
In arithmetic, the order of operations is a set of rules used to determine which operations to perform first in a mathematical expression. This is essential for ensuring consistent results.
The order of operations can be remembered by the acronym PEMDAS:
By systematically following these rules, you can accurately simplify even complex arithmetic expressions.
The order of operations can be remembered by the acronym PEMDAS:
- Parentheses
- Exponents
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
By systematically following these rules, you can accurately simplify even complex arithmetic expressions.
