Question

Without a calculator, decide whether the quantities are positive or negative. $$ (-4)^{3} $$

Short answer

Answer: The result of the quantity (-4)^3 is -64, which is a negative number.

Step-by-step solution

Understand the expression

We have an expression of the form \((-4)^{3}\). This means we need to multiply -4 by itself three times: \((-4) \times (-4) \times (-4)\).

Multiply the first two numbers

First, let's multiply the first two instances of -4: \((-4) \times (-4) = 16\). Multiplying two negative numbers results in a positive number.

Multiply the result by the last number

Now, we multiply the result from step 2 by the last instance of -4: \(16 \times (-4) = -64\). Multiplying a positive number by a negative number results in a negative number.

Determine the sign of the result

From step 3, we have the result as \((-4)^{3} = -64\). Since -64 is a negative number, we can conclude that the quantity is negative.

Key concepts

Multiplication of Integers

When working with integers, particularly in algebraic expressions, understanding multiplication is crucial. Integers include both positive and negative whole numbers, as well as zero. Multiplying integers, especially those with opposite signs, can seem tricky at first.
To simplify the process:

• Two positive numbers multiplied together yield a positive result.
• Two negative numbers multiplied together also give a positive result.
• A positive number and a negative number multiplied together result in a negative product.

In the exercise, initially, \((-4) \times (-4) = 16\) is calculated, illustrating multiplication between two negative integers, resulting in a positive product. This foundational rule helps set the stage for more complex calculations, such as those involving exponents.

Negative Numbers

Negative numbers can often cause confusion, especially in the realm of multiplication or exponentiation. These numbers are less than zero and are represented with a minus sign. When dealing with negative numbers:
It's essential to recognize:

• Multiplying two negatives, as earlier noted, turns the result positive since the two negatives cancel each other out.
• On the other hand, multiplying a negative number by a positive retains the negative sign, indicating that one negative remains.

Applying these concepts is crucial, as seen in the previous calculation steps of the exercise. Recognizing how each operation affects the sign of the product prepares you to manage more complex expressions smoothly.

Exponentiation

Exponentiation is a shorthand way of expressing repeated multiplication of the same number. An exponent indicates how many times the base (the number being multiplied) appears in the expression.
For example, in \((-4)^3\), -4 is the base, and 3 is the exponent, meaning that -4 is multiplied by itself three times:

• \((-4) \times (-4) \times (-4)\)

This becomes a series of integer multiplications, each influenced by the rules for multiplying integers. As demonstrated, the first multiplication of two negatives gives a positive, which then multiplied with another negative yields a negative outcome. Understanding exponentiation and its effect in expressions involving negative numbers ensures accuracy in problem-solving and prepares one for tackling varied algebraic problems.