Question

For Exercises \(1-74,\) simplify. $$-|38-14 \cdot 2|+44 \div[5-(-6)]-2^{4}$$

Short answer

Answer: The simplified form of the expression is -22.

Step-by-step solution

Simplify the expression within the absolute value function

Start by performing the operation inside the absolute value function first: $$|-|38 - 14\cdot2|$$ $$|-|38 - 28|$$ $$|-|10|$$

Evaluate the absolute value function

Now, evaluate the absolute value of 10: $$|-10|$$ $$-10$$

Simplify the expression within the brackets

Move on to simplifying the expression in the brackets: $$44\div [5 -(-6)]$$ $$44\div [5 + 6]$$ $$44\div 11$$

Evaluate the division

Now, evaluate the division: $$44\;\div\; 11 = 4$$

Evaluate the exponent

Next, solve the exponent: $$2^4$$ $$16$$

Combine the results from steps 2, 4, and 5

We now have: $$-10 + 4 - 16$$

Evaluate the expression from left to right

Perform the addition and subtraction from left to right: $$-6 - 16 = -22$$ The simplified expression is \(-22\).

Key concepts

Absolute Value

The concept of absolute value reflects the distance of a number from zero on the number line, regardless of its direction. It is represented by two vertical bars surrounding the number, for example, \( |x| \). To simplify an expression with an absolute value, you need to first remove the absolute value by considering the sign inside it.

If you have a negative number inside, such as \( |-10| \), you convert it to \( 10 \), because the absolute value of a negative number is the positive of that number. Consequently, in the exercise \( -|38-14 \cdot 2| \), after calculating the expression within the absolute value to get \( |-10| \), the absolute value simplifies to \( 10 \. Remember that the negative sign in front of the absolute value remains, making the whole expression \( -10 \).

Order of Operations

Mathematical expressions are not just read from left to right. Instead, they are simplified by following a specific order known as the order of operations. This is often abbreviated as PEMDAS or BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

When simplifying an expression like \( -|38-14 \cdot 2|+44 \div[5-(-6)]-2^{4} \), you must start by solving any operations inside parentheses or brackets. Next, handle exponents, followed by any multiplication or division from left to right, and finally, addition and subtraction from left to right. In our exercise, these rules lead us step by step through complex operations within absolute values and brackets, to straightforward multiplication, division, and exponentiation.

Exponentiation

Exponentiation is an operation that raises a number to a power. It's represented with a smaller numeral above and to the right of a base number, such as \( a^b \), where \( b \) is the exponent and \( a \) is the base. It means that \( a \) is multiplied by itself \( b \) times. For example, \( 2^4 \) means \( 2 \times 2 \times 2 \times 2 \), which is 16.

In the given exercise, after calculating the expression within the absolute value and the divisions, the exponent \( 2^4 \) was calculated as the next step. This is critical, as according to the order of operations, exponents must be handled before any multiplication, division, or addition/subtraction processes (unless these last operations are within brackets or parentheses).