Question

Without a calculator, decide whether the quantities are positive or negative. $$ (-47)^{-15} $$

Short answer

Answer: The sign of the expression \((-47)^{-15}\) is negative.

Step-by-step solution

Recognize the negative exponent

A negative exponent means we need to take the reciprocal of the base number. Remember that the reciprocal of a number is simply 1 divided by that number. In this case, our base number is -47, so we write the expression as: $$ \frac{1}{(-47)^{15}} $$

Identify the effect of the odd exponent

Now we need to determine what happens to the sign of the base number when raised to an odd exponent. When a negative number is raised to an odd power, the result will be negative. This is because multiplying an odd number of negative numbers will result in a negative product. In our case, we have: $$ (-47)^{15} = -(47^{15}) $$

Combine the previous steps

Now let's substitute our result from step 2 back into the expression from step 1: $$ \frac{1}{(-47)^{15}} = \frac{1}{-(47^{15})} $$

Determine the sign of the final result

We now have a positive number (1) divided by a negative number (\(-(47^{15})\)). When a positive number is divided by a negative number, the result is negative. So, the value of \((-47)^{-15}\) is negative.

Key concepts

Reciprocal of a Number

When working with negative exponents, you often come across the term "reciprocal". The reciprocal of a number is what you multiply that number by to get 1. For example, the reciprocal of 5 is \( \frac{1}{5} \), because \( 5 \times \frac{1}{5} = 1 \).

When dealing with negative exponents, the idea of reciprocals becomes very useful. A negative exponent tells you to take the reciprocal of the base and change the sign of the exponent to positive. Thus, any number \(a^{-b}\) can be written as \(\frac{1}{a^{b}}\).

For instance, in the exercise, \((-47)^{-15}\) becomes \(\frac{1}{(-47)^{15}}\), because

• you move the base from the numerator to the denominator, effectively taking its reciprocal
• you change the exponent from negative to positive

This helps simplify calculations and is key in most algebraic manipulations involving negative exponents.

Odd Exponents Impact

Exponents tell you how many times to multiply a number by itself. An odd exponent, like 3 or 15, plays a specific role in determining the sign of the result, especially with negative bases.

When you raise a negative number to an odd exponent, the result is always negative. This is because each pair of negative numbers multiplied together becomes positive, and any leftover unpaired negative will make the overall product negative.

So, for \((-47)^{15}\), since 15 is an odd number, you multiply \(-47\) by itself 15 times. Each pair of \(-47 \times -47\) turns positive, but you end up with one extra \(-47\), ensuring the entire product is negative. Specifically,

• each combination of two negatives results in a positive
• the odd leftover keeps the entire output negative

This understanding is crucial when figuring out the effects of any exponent on a negative base.

Sign of a Product

Understanding how signs affect multiplication can help discern the final sign of a product. This is especially important when dealing with expressions like \((-47)^{-15}\). Here's a simple rule:

• Multiplying two negative numbers results in a positive product.
• Multiplying a positive number by a negative number results in a negative product.

In the context of our example, we analyze the fraction \(\frac{1}{-(47^{15})}\). We have a positive numerator (which is 1) and a negative denominator (because the odd exponent applied to -47 results in a negative number).

When dividing:

• A positive divided by a positive remains positive.
• A positive divided by a negative turns negative.

This means that the product or final result of dividing a positive number by a negative number, as in our exercise, is inevitably negative. This basic principle helps ensure accuracy in calculations without needing actual numerical computation.