Question

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

Short answer

Answer: The result of the expression \((-3)^{4}\) is positive.

Step-by-step solution

Identify the base and exponent

The expression \((-3)^{4}\) consists of a base -3 and exponent 4. Our task is to find whether the answer is positive or negative by analyzing the base (-3) and the exponent (4).

Evaluate the expression using the exponent properties

In this step, we will use the property of the exponent that states that a negative number raised to an even integer power results in a positive number. Here, the base (-3) is a negative number and the exponent 4 is an even integer. Thus, \((-3)^{4}\) will result in a positive number.

Conclude the result

As per the exponent properties, \((-3)^{4}\) will be a positive number.

Key concepts

Negative Base

When working with exponents, it's important to closely look at the base number, especially if it's negative. A negative base can significantly affect the outcome depending on whether the exponent is even or odd.
A negative base simply means a number that is less than zero. When this base is only enclosed in parentheses as in \((-3)^4\), it implies that both the negative sign and the number are included in the base being raised to the power. So here, \(-3\) is the complete base, meaning we multiply \(-3\) by itself four times. If it was written without parentheses, such as \-3^4\, the negative would only apply once and we are only raising 3 to the fourth power, \(-1 \times 3^4\).
In short:

• Parentheses around a negative base mean both the sign and the number are raised to the power.
• No parentheses means only the number is subject to the exponentiation, and then the negative is applied.

Even Exponent

The role of the exponent in an expression is to determine how many times the base is multiplied by itself. If the exponent is an even number, interesting things happen, especially with negative bases.
An even exponent transforms any negative base into a positive number. This happens because multiplication by a negative number an even number of times results in a positive outcome. Each multiplication switches the sign, so pairs of negative terms cancel out their negativity, leading to a positive result.
So, when you see \((-3)^4\), remember these key points:

• Even number of multiplications (e.g., 2, 4, 6) makes a negative base positive.
• Odd exponents result in a negative number unless the base is positive.

This trick comes from recognizing the repeated multiplications inherent in exponentiation.

Exponent Properties

Underlying most exponent operations are basic properties that dictate how numbers behave. These properties save us from having to do all the multiplications manually and help predict results quickly. One such property is that any negative number taken to an even power results in a positive number. This is because of what happens during the process of continuous multiplication, alternating signs till they end positive.
Key properties include:

• Negative numbers with even exponents yield positive results.
• Negative numbers with odd exponents yield negative results.
• Any non-zero number to the power of zero is \(1\).

These principles allow efficient calculations and correct conclusions, like determining quickly that \((-3)^4\) must be positive because the exponent is even. By familiarizing yourself with these properties, solving exponent questions becomes a lot easier and more intuitive.