Problem 32 - Prealgebra and Introductory Algebra

Question

Simplify. $$5-2\left|(-3)^{2}-8\right|$$

Step-by-Step Solution

Answer

The simplified form of the given expression is 3.

Step 1: Solve the Exponent

Apply the exponent operation first. \((-3)^2\) equals to \(9\). The equation now reads \(5 - 2\left|9-8\right|\).

Step 2: Solve inside the absolute value brackets

Subtract the numbers inside the absolute value brackets. The absolute value of \(9-8\) is \(1\). Therefore, the equation reads \(5 - 2*1\).

Step 3: Perform Multiplication

Multiply \(2\) with the absolute value, which is \(1\). This results in \(2\). The equation now reads \(5-2\).

Step 4: Perform Subtraction

Subtract \(2\) from \(5\). This equals to \(3\).

Key Concepts

Understanding Exponents

Exponents represent a number multiplied by itself a specified number of times. Here’s how they work: when you see a number like \( (-3)^2 \), it simply means \( (-3) \times (-3) \). This equals \( 9 \). The exponent '2' tells us how many times we use \((-3) \)as a factor.

A few quick tips about exponents:

• When you raise a negative number to an even exponent, the result is positive. For example, \( (-3)^2 = 9 \).
• When you raise a negative number to an odd exponent, the result is negative. For example, \( (-3)^3 = -27 \).
• Any number raised to the exponent '0' is always \( 1 \), except for zero itself.

Using these principles, simplify expressions containing exponents efficiently and correctly.

Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering direction. It’s always a positive number or zero.

In the expression \( |-3| \), the absolute value is \( 3 \), because \( 3 \)and \(-3 \)both have the same distance (3 units) from zero.

• The absolute value of a positive number, like \( |7| \), remains \( 7 \).
• The absolute value of zero, \( |0| \), is \( 0 \).
• For negative numbers, such as \( |-5| \), you take away the negative sign for an absolute value of \( 5 \).

Absolute values are helpful in calculations where only the magnitude of a number is important, such as in the expression\( 5 - 2\left|9-8\right| \), where the absolute value simplifies calculations by providing a straightforward positive result.

Order of Operations Simplified

The order of operations is a set of rules that tells us the sequence to follow when solving a math expression to ensure everyone arrives at the same answer.

The acronym PEMDAS helps remember the order:

• Parentheses first
• Exponents (i.e., powers and roots, etc.)
• Multiplication and Division (left-to-right)
• Addition and Subtraction (left-to-right)

By following PEMDAS:- First, simplify anything inside parentheses or absolute value bars.- Next, solve exponents.- Then, perform multiplication or division, moving left to right across the equation.- Finally, handle addition or subtraction, again working left to right.

In the problem \( 5 - 2\left|(-3)^2-8\right| \), we first manage the exponent, then address what's inside the absolute value, multiply next, and ultimately complete the subtraction. This structured approach ensures accurate results every time.