Question

Without a calculator, decide whether the quantities are positive or negative. $$ (-23)^{42} $$

Short answer

Answer: The result of the expression \((-23)^{42}\) is positive.

Step-by-step solution

Identify the base and the exponent

The expression in question is \((-23)^{42}\). Here, the base is -23, which is a negative number, and the exponent is 42, which is an even integer.

Determine the result of raising a negative base to an even exponent

When a negative base is raised to an even exponent, the result will always be positive. This is because an even exponent can be represented as \(2n\), where \(n\) is an integer. Therefore, the expression becomes: $$ (-a)^{2n} = (-a) * (-a) * ... * (-a) \ [2n \ times] $$ As there are an even number of negative signs, they will all be paired up and result in a positive value.

Apply the result to the given expression

Since we know that a negative base raised to an even exponent results in a positive value, we can apply this knowledge to the given expression \((-23)^{42}\). As 42 is an even integer, the result will be positive.

State the final answer

The final answer is that the expression \((-23)^{42}\) is positive.

Key concepts

Even Exponent

When working with exponents, particularly negative bases, noticing whether the exponent is even or odd is crucial. An even exponent means that the power is expressed as a multiple of two, such as 2, 4, 6, 8, and so on.

Even exponents have a unique property, especially when dealing with negative numbers:

• The even-numbered power of any negative number will result in a positive number.
• This happens because multiplying two negative numbers together yields a positive product.

For example, consider the expression \[ (-a)^{2n} \]. Here, the base -a is squared, cubed, or raised to any even number of times, ensuring pairs of negative numbers cancel out their signs. This concept helps decide the outcome's sign, irrespective of the base's value. Another handy point to remember is that an even exponent does not change the absolute size of the original base, just the sign of its outcome.

Positive Result

Understanding the outcome when raising a negative base to an even exponent relies heavily on knowing that the result will always be positive.

Why? Let's dive into the logic:

• Each negative factor in the base has a corresponding partner due to the even exponent.
• When these negative pairs multiply, they always produce a positive product, which leads to an overall positive result for the entire expression.

By examining expressions like \[ (-3)^6 = (-3) imes (-3) imes (-3) imes (-3) imes (-3) imes (-3) \], we can see that each negative pair (\[ (-3) imes (-3) \], for example) results in +9. Continue pairing until the power is exhausted, revealing that negative signs simply "cancel each other out," resulting in a positive number. Even though the base number is negative, knowing the exponent's characteristics allows us to predict the result accurately without detailed calculations.

Integer Powers

Exploring integer powers involves understanding how they change or maintain certain base characteristics. Powers can work seamlessly with both negative and positive bases.

• Integer powers include even powers, odd powers, and zero as an exponent.
• Raising a base to a positive integer power involves multiplying the base by itself repeatedly.

For example, consider the expression \[ b^n \], where "b" is a base and "n" is a non-zero integer. This signifies the base "b" multiplied by itself "n" times.

However, note the uniqueness of even powers in making a negative base yield a positive result, as we've seen above. Odd powers retain the negative sign in such situations because there is one negative factor left after pairing, leading to a negative product. This understanding is intrinsic to manipulating mathematical expressions and gaining fluency in exponentiation. Regardless of the storyline that integer powers tell, their simplified calculations usually involve patterns that are manageable with practice.