Problem 18 - Algebra A Combined Function

Question

Evaluate \((-3)^{3}\)

Step-by-Step Solution

Answer

The result of \((-3)^3\) is \(-27\).

Step 1: Identify the Base and the Exponent

The expression given is \((-3)^3\). Here, the base is the number being raised to a power, which is -3, and the exponent is 3. This means we need to multiply the base \(-3\) by itself a total of 3 times.

Step 2: Multiply the Base by Itself Repeatedly

Start by multiplying the base \(-3\) by itself: \((-3) \times (-3) = 9\)Since the exponent is 3, we need one more multiplication.

Step 3: Multiply the Result by the Base Again

Now take the result from Step 2, which is 9, and multiply by the base \(-3\) again:\(9 \times (-3) = -27\)

Step 4: Conclude the Calculation

The result of \((-3)^3\) is \(-27\). This is the final answer, as we have performed all necessary multiplications according to the exponent.

Key Concepts

Understanding Negative Numbers

Negative numbers follow the same mathematical rules as positive numbers but have the unique property of being less than zero. Working with negative numbers requires careful consideration, especially during multiplication or division. When multiplying two negative numbers, the result is positive because the negatives cancel each other out. For example,

• If you multiply \(-3\times-3\), you get \(9\).

However, when you multiply a negative number by a positive number, as with \(9 imes-3\), the result is still negative. That's because there's essentially more negative influence than positive. Here, the rules of negative numbers are crucial in determining the result of the exponentiation. When raising \(-3\) to the power of 3, you need to perform multiple multiplications, carefully respecting these sign rules.

Exploring Powers of Integers

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. When dealing with powers of integers, you raise the base integer to the strength of the exponent by multiplying the base by itself several times. For instance, in the expression \((-3)^3\),

• The base is \(-3\), and the exponent is 3.
• This means you perform the following operation: \((-3) imes (-3) imes (-3)\).

The base remains consistent while the multiplication goes as many times as the exponent dictates. With integer exponents, especially odd ones, if the base is negative, the ultimate result is negative, while evens yield positive results. Hence, \((-3)^3 = -27\).Exponential operations can sometimes be confusing, but they simplify the multiplication process significantly by structuring it into smaller sequential steps.

Understanding Multiplication Sequences

A multiplication sequence involves multiplying numbers in a specific order to achieve the desired product. In exponentiation such as \((-3)^3\), a clear sequence must be followed. You first multiply the base by itself to the extent of the exponent. Here's how it works step-by-step:

• First, multiply \(-3\times-3\) to get \(9\).
• Then, multiply \(9\times-3\)

to reach the final product of \(-27\).Each multiplication step must respect the sequence, as set by the exponent, to achieve an accurate calculation. This ordered process ensures the final result aligns with mathematical principles, particularly the rules about the signs of numbers. Recognizing the multiplication sequence as a series of small multiplications ensures clarity in understanding exponentiation.