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Chapter 1: Problem 15

Evaluate each expression when \(x=-2, y=7,\) and \(z=-3\) $$z^{3}-x^{3}$$

Short Answer

Expert verified
The evaluated expression when x = -2, y = 7, and z = -3 is: \(-35\)

Step by step solution

01

Write down the expression and given values of x, y, and z

We will evaluate the expression: \(z^3 - x^3\) with the values: x = -2, y = 7, and z = -3. Note: The value of y is not used in this expression.
02

Substitute the values of x and z in the expression

Plug in the values of x and z in the expression: \((-3)^3 - (-2)^3\)
03

Calculate the powers of x and z

Now, calculate the powers of 3 for both x and z: \((-27) - (8)\)
04

Simplify the expression

Now, simplify the expression by performing the subtraction: \(-27 - 8 = -35\) So, the evaluated expression when x = -2, y = 7, and z = -3 is: \(-35\)

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Key Concepts

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

Substitution Method
When solving algebraic expressions, often the first step is to replace the variables with specific numbers. This process is known as the substitution method.

Let's break it down: you take each variable鈥攍ike \(x\), \(y\), or \(z\)鈥攁nd substitute it with its given value.
  • In the example, you were given: \(x = -2\), \(y = 7\), and \(z = -3\). However, only \(x\) and \(z\) were used in the expression \(z^3 - x^3\).
  • This means we substitute \(z\) with \(-3\) and \(x\) with \(-2\).
So whenever you're asked to "evaluate an expression" using specific values, remember: substituting each variable with the given numbers is your first move.
Evaluating Expressions
Evaluating expressions is simply a way to find out the value of an algebraic expression after substituting its variables with numbers. It involves performing basic arithmetic operations.

For our particular exercise:
  • Once we substitute \(z\) with \(-3\) and \(x\) with \(-2\), the expression becomes \((-3)^3 - (-2)^3\).
  • The next step is to calculate each power separately: \((-3)^3 = -27\) and \((-2)^3 = -8\).
Then, you're left with a straightforward calculation: \(-27 - 8\). This operation results in \(-35\).

If evaluating expressions seems tricky, just remember to take it step by step, substituting values and performing arithmetic operations wisely.
Integer Exponents
Integer exponents refer to raising a number to a particular power, which indicates repeated multiplication of that number by itself. Understanding integer exponents simplifies many algebraic expressions and calculations.

For the expression \(z^3 - x^3\), we used integer exponents on both \(z\) and \(x\):
  • Raising \(z = -3\) to the power of 3: \((-3)^3 = -3 \times -3 \times -3 = -27\). Negative multiplied by negative gives a positive, but multiplied again by a negative, results in a negative.
  • Similarly, raising \(x = -2\) to the power of 3: \((-2)^3 = -2 \times -2 \times -2 = -8\).
Integer exponents can involve positive and negative numbers, resulting in positive or negative products.

With exponents, it's essential to carefully calculate the sign based on the number of times your base number is multiplied, especially with negative bases.

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Most popular questions from this chapter

Write a mathematical expression for each and simplify. The quotient of \(-100\) and 4 decreased by the sum of \(-7\) and 2

Use the order of operations to simplify. $$-7(4)+(-8+6)^{4}+5$$

Use the order of operations to simplify. $$\left(-\frac{3}{4}\right)(8)-2[7-(-3)(-6)]$$

Rewrite each expression using the distributive property. Simplify if possible. $$12(2 a-3 b+c)$$

Applications of Perimeter, Area, and Volume: Use 3.14 for \(\pi\) and include the correct units. A pile of sand is in the shape of a right circular cone. The radius of the base is \(2 \mathrm{ft}\), and the pile is \(6 \mathrm{ft}\) high. Find the volume of sand in the pile.
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