Problem 36 - Algebra 1

Question

Simplify the variable expression. $$|(8)(-z)(-z)(-z)|$$

Step-by-Step Solution

Answer

The simplification gives \( |-8z^3| \), but its value depends on \( z \). If \( z \geq 0\), the result is \( 8z^3\); If \( z < 0\), the result is \(-8z^3\). It cannot be simplified further without knowing the value of \( z \).

Step 1: Perform the multiplication.

To start, perform the multiplication inside the absolute value bars. The multiplication gives \( (8)(-z)(-z)(-z) = -8z^3\).

Step 2: Simplify using absolute value.

Next, apply the absolute value operation to \(-8z^3\). In mathematical terms, the absolute value of a number is its distance from zero and is always non-negative. Therefore \( |-8z^3| = 8z^3\) if \( z \geq 0\) and \( |-8z^3| = -8z^3\) if \( z < 0 \). However, we do not know the value of \( z \), so we cannot simplify it further.

Key Concepts

Understanding Absolute Value

When simplifying expressions, it is crucial to understand what absolute value represents. The absolute value of a number is a measure of the distance of that number from zero on the number line, regardless of direction. It is always positive or zero because distance cannot be negative. In mathematical notation, the absolute value of a number 'x' is represented as \( |x| \).

For example, the absolute value of -5 is 5 (\(|-5| = 5\)), and the absolute value of 5 is also 5 (\(|5| = 5\)). In the context of the exercise provided, applying the absolute value to a variable expression involving negative numbers, such as \( |-8z^3| \), results in the negative sign being 'stripped off,' leaving us with a positive \( 8z^3 \) if \( z \) is a real number. With variables, unless we know if \( z \) is positive or negative, we cannot determine a further simplified form beyond stating that the result will be non-negative.

Multiplying Variables

When working with algebraic expressions, multiplying variables is a common operation. The general rule for multiplying variables is straightforward: when you multiply two variables of the same base, you simply add their exponents. This is an application of the property of exponents stating that \( a^m \cdot a^n = a^{m+n} \).

In the exercise, we are dealing with the multiplication of the same variable \( z \) three times, coupled with the multiplication by 8. This is represented as \( (8)(-z)(-z)(-z) \) which simplifies to \( -8z^3 \) by multiplying the coefficient (8) by the repeated multiplication of \( z \) (raised to the power of 3 due to there being three instances of \( z \) being multiplied together). Remember, the negative sign arises from the fact that there is an odd number of negative factors being multiplied.

Exponents and Their Properties

Exponents play a crucial role in simplifying mathematical expressions. An exponent indicates how many times a number, known as the base, is to be multiplied by itself. For instance, \( x^3 \) means \( x \) multiplied by itself three times (\( x \times x \times x \)).

Understanding the properties of exponents can significantly simplify the process of working with them. Key properties include the product of powers rule (\( a^m \times a^n = a^{m+n} \)), the power of a power rule (\( (a^m)^n = a^{mn} \)), and the power of a product rule (\( (ab)^n = a^n b^n \)), among others. In our exercise, the expression \( -8z^3 \) contains a power of \( z \) which was derived from multiplying \( z \) by itself three times. To simplify expressions with exponents, always combine like terms and apply the relevant exponent rules accurately.