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Chapter 4: Problem 29

Perform each indicated operation. See Section \(1.3 .\) $$ (-2)^{2}-(-3)+2(-1) $$

Short Answer

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Step by step solution

01

Evaluate the Exponent

Begin by solving the exponent in the expression. Here, \[(-2)^2 = (-2) imes (-2) = 4.\]Replace \((-2)^2\) in the expression with \(4\).
02

Simplify the Subtraction of Negative

Next, simplify the expression by solving \(-(-3)\). Subtracting a negative is equivalent to adding a positive: \[-(-3) = +3.\]Replace \(-(-3)\) in the expression with \(+3\).
03

Multiply the Terms

Now, calculate the multiplication in the expression: \[2(-1) = -2.\]Replace \(2(-1)\) in the expression with \(-2\).
04

Combine All Results

Now, substitute each evaluated part back into the original expression and perform the operations: \[4 + 3 - 2.\]Add and subtract in order from left to right: \[4 + 3 = 7,\]and then,\[7 - 2 = 5.\]

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Key Concepts

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

Understanding Exponents
Exponents are an essential mathematical concept that describe a number raised to a power. In simple terms, an exponent tells you how many times to multiply a number by itself. When you see something like \((-2)^2\), the number \(-2\) is the base, and \(2\) is the exponent. This means you should multiply \(-2\) by itself: \((-2) \times (-2) = 4\). This result, \(4\), is what replaces \((-2)^2\) in the original expression.
  • Exponents simplify long multiplication
  • Always focus on the sign of the base when raising it to a power
  • Even exponents of negative numbers lead to positive results due to the property of multiplication between two negative numbers
This understanding helps you efficiently evaluate parts of complex expressions containing exponents.
Handling Negative Numbers
Dealing with negative numbers can be tricky, but they are crucial in math. A key to understanding them is realizing that negative numbers are just numbers on the opposite side of zero on the number line.
When you encounter a subtraction preceded by a negative sign, like with \(-(-3)\), you transform it into an addition owing to the rule that subtracting a negative is equivalent to adding a positive:
  • Negative signs can change the operation in an expression
  • Two negative signs interacting turn into a positive
  • Think of subtracting a negative as a double negative, which makes a positive
This conversion of signs is crucial to prevent mistakes in calculations to get the correct result.
Mastering Multiplicative Operations
Multiplicative operations involve calculating products between numbers. In algebraic expressions, such operations may involve negative signs, altering the results. For instance, multiplying a positive number by a negative number, like \(2(-1)\), results in a negative product: \(2 \times (-1) = -2\).
  • Always consider the signs of numbers involved in multiplication
  • The product of a positive and a negative number is negative
  • Rows of products form when dealing with multiple multiplied terms
Grasping how multiplicative operations alter the result ensures precision when simplifying algebraic expressions.

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Most popular questions from this chapter

Solve each system of equations by the elimination method. See Examples 7 through 10. $$ \left\\{\begin{array}{l} {\frac{2}{3} x+\frac{1}{4} y=-\frac{3}{2}} \\ {\frac{1}{2} x-\frac{1}{4} y=-2} \end{array}\right. $$

Solve each system by graphing. See Examples 2, through 4. $$ \left\\{\begin{array}{l} {3 x-y=4} \\ {6 x-2 y=4} \end{array}\right. $$

Solve each system of equations by the substitution method. See Examples 5 and 6. $$ \left\\{\begin{array}{c} {\frac{x}{3}+y=\frac{4}{3}} \\ {-x+2 y=11} \end{array}\right. $$

Determine whether the given replacement values make equation true or false. See Section 1.3. $$ x+2 y-z=7 ; x=2, y=-3, \text { and } z=3 $$

Solve each system of equations. $$ \left\\{\begin{array}{c} {\frac{x+5}{2}=\frac{6-4 y}{3}} \\ {\frac{3 x}{5}=\frac{21-7 y}{10}} \end{array}\right. $$
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