Problem 50 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$3(-3)^{3}+4(-3)^{2}-5(-3)+7$$

Step-by-Step Solution

Answer

The simplified expression is -23.

Step 1: Calculate \((-3)^{3}\)

First, calculate the cube of \(-3\): \[ (-3)^{3} = (-3) imes (-3) imes (-3) = -27 \]

Step 2: Multiply \(3\) and \((-3)^{3}\)

Now that we have \((-3)^{3} = -27\), multiply it by 3: \[ 3(-3)^{3} = 3 imes (-27) = -81 \]

Step 3: Calculate \((-3)^{2}\)

Calculate the square of \(-3\): \[ (-3)^{2} = (-3) imes (-3) = 9 \]

Step 4: Multiply \(4\) and \((-3)^{2}\)

With \((-3)^{2} = 9\), multiply it by 4: \[ 4(-3)^{2} = 4 imes 9 = 36 \]

Step 5: Calculate \(-5(-3)\)

Next, multiply \(-5\) and \(-3\): \[-5(-3) = 15\]

Step 6: Add up all computed values

Finally, combine all the computed values:\[ -81 + 36 + 15 + 7 \]Start by adding from left to right:\[ (-81 + 36) = -45 \]\[ (-45 + 15) = -30 \]\[ (-30 + 7) = -23 \]

Key Concepts

Order of Operations

When simplifying numerical expressions, the order of operations is a crucial concept to understand. In mathematics, there is a specific sequence in which operations should be performed to ensure consistent and correct results. This order is sometimes remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

It is essential always to perform operations inside parentheses first. Following that, handle the exponents. Afterward, execute multiplication and division from left to right, and finally, address addition and subtraction, also from left to right. By adhering to this sequence, it ensures that complex expressions are simplified without error.

Exponents

Exponents are a mathematical notation indicating the number of times a number, known as the base, is multiplied by itself. For example, in \((-3)^3\), \(-3\) is the base and \(3\) is the exponent. This means you multiply \(-3\) by itself three times: \((-3) \times (-3) \times (-3) = -27\).

Understanding exponents is critical in simplifying expressions involving powers. It's helpful to remember that an even exponent results in a positive outcome if the base is negative, like \((-3)^2 = 9\). In contrast, an odd exponent retains the negative sign as in \((-3)^3 = -27\).

It's important to handle exponents before tackling multiplication or other operations to make sure the numbers are simplified correctly as per the order of operations.

Basic Arithmetic Operations

Basic arithmetic operations are the building blocks of all mathematical calculations. They include addition, subtraction, multiplication, and division.

• Addition combines two or more numbers to get a sum.
• Subtraction finds the difference between numbers by taking one away from another.
• Multiplication is repeated addition, where a number is added to itself a specified number of times.
• Division partitions a number into equal parts. It's the inverse of multiplication.

When simplifying a complex expression, multiply before adding and subtracting when following the order of operations. Having a strong grasp of these operations is essential for accurate and efficient problem-solving.

Integer Arithmetic

Integer arithmetic concerns the operations involving whole numbers, both positive and negative. Integers do not include fractions or decimals, and they are crucial in various fields of mathematics.

In our expression \(3(-3)^3 + 4(-3)^2 - 5(-3) + 7\), integer arithmetic helps in performing calculations such as \(-81 + 36 + 15 + 7\).

The rules of adding, subtracting, multiplying, and dividing integers are:

• **Addition**: When adding integers with the same sign, add their absolute values and keep the sign. When adding integers with different signs, subtract the smaller absolute value from the larger absolute value, and use the sign of the larger absolute value.
• **Subtraction**: Convert the subtraction of integers into the addition of a negative and then apply the addition rules.
• **Multiplication and Division**: If the signs are the same, the answer is positive. If the signs are different, the answer is negative.

Understanding these rules ensures the correct simplification of expressions involving integers.