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Chapter 2: Problem 76

Evaluate each expression for \(x=-3\) and \(z=-4\) $$ 3-z^{2} $$

Short Answer

Expert verified
The value of the expression is -13.

Step by step solution

01

Substitution of Given Values

Substitute the given value of \(z = -4\) into the expression \(3 - z^{2}\). This gives us \(3 - (-4)^{2}\).
02

Calculate the Square of z

Calculate \((-4)^{2}\). The result is 16 because any negative number squared becomes positive. Thus, \((-4)^{2} = 16\).
03

Substitute the Squared Value

Replace \((-4)^{2}\) with 16 in the expression. Now the expression is \(3 - 16\).
04

Final Calculation

Perform the subtraction \(3 - 16\). This gives \(-13\).

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Key Concepts

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

Negative Numbers
Negative numbers are numbers that are less than zero. They have a minus sign (-) in front of them. Understanding how negative numbers work is crucial when performing mathematical operations like addition, subtraction, multiplication, and division.
  • When you add a negative number, it is like subtracting the absolute value of that number.
  • Subtracting a negative number is equivalent to adding the positive version of that number.
  • When you multiply or divide two negative numbers, the result is positive.
  • However, multiplying or dividing a negative number with a positive number results in a negative number.
In the given exercise, we substituted the value of \( z = -4 \), which is a negative number. Notice how the negative sign is critical when working with these values, especially in operations like squaring.
Exponents
Exponents are a way to express repeated multiplication. For instance, \( z^2 \) means \( z \) multiplied by itself. If \( z \) is negative, you need to be cautious about the result.
  • If the exponent is even, the result is always positive, regardless of whether the base is positive or negative. For example, \((-4)^2\) gives 16.
  • If the exponent is odd, the result keeps the sign of the base. For instance, \((-4)^3 = -64\).
In our exercise, when calculating \((-4)^2\), the negative sign goes away because the square of a negative number is positive. This is why it is essential to pay attention to the exponent when working with negative numbers.
Evaluating Expressions
Evaluating expressions is all about finding the value of the expression by substituting the variables with given numbers. This involves a careful execution of arithmetic operations according to mathematical rules:
  • Start by substituting the values of the variables into the expression.
  • Follow the order of operations鈥攑arentheses, exponents, multiplication and division, and addition and subtraction (often remembered by PEMDAS or BODMAS).
  • Perform each calculation step-by-step to avoid any mistakes.
In the provided solution, we begin by substituting \( z = -4 \) into the expression \( 3 - z^2 \), yielding \( 3 - (-4)^2 \). Then, calculating \((-4)^2\) gives 16, changing the expression to \( 3 - 16 \). Finally, solving \( 3 - 16 \) results in \(-13\), which is the evaluated expression's value. As you can see, each step builds upon the previous, clearly illustrating the method of evaluating expressions.

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Most popular questions from this chapter

Represent each quantity by an integer. Uber lost \(\$ 1270\) million in the first half of 2016 .

Insert \(<,>,\) or \(=\) between each pair of numbers to make a true statement. $$ -22 \quad-(-38) $$

Represent each quantity by an integer. A scuba diver is swimming 25 feet below the surface of the water in the Gulf of Mexico.

Represent each quantity by an integer. The Challenger Deep in the Mariana Trench (the deepest-known part of the Pacific Ocean) is the lowest known point in the Earth's seabed hydrosphere. The bottom is estimated to be 35,814 feet below sea level.

Add. $$ 0+13 $$
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