Problem 15 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. $$-81^{0.5}$$

Step-by-Step Solution

Answer

The expression evaluates to -9.

Step 1: Assess the Expression

The expression given is \(-81^{0.5}\). This can be rewritten using a root form, where the exponent \(0.5\) corresponds to the square root. Therefore, it translates to \(-\sqrt{81}\).

Step 2: Calculate the Square Root

Calculate the square root of 81, which is 9, since \(9 \times 9 = 81\).

Step 3: Apply Negation

The problem expression is \(-\sqrt{81}\), which means you need to apply the negative sign to the result obtained from Step 2. Therefore, \(-9\) is the result.

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. For example, the exponent 2 in the term \(3^2\) signifies that 3 is multiplied by itself once (i.e., \(3 \times 3\)). The result is 9. An exponent is also known as a power. More generally, for any number \(a\) raised to a power \(n\), written as \(a^n\), means multiplying \(a\) by itself \(n-1\) times:

• \(a^1 = a\)
• \(a^2 = a \times a\)
• \(a^3 = a \times a \times a\)

The interesting part about exponents is how they behave with fractions, like in the expression \(81^{0.5}\). The exponent 0.5 is actually another way to express finding a square root. Thus, \(81^{0.5}\) is equivalent to \(\sqrt{81}\). This is a useful representation as it connects the concepts of roots and powers seamlessly, offering us a powerful tool in simplifying expressions.

Diving into Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. The square root is represented by the radical symbol, \(\sqrt{}\). For instance, \(\sqrt{81} = 9\) because \(9 \times 9 = 81\). Square roots are a form of inverse operation to squaring a number (exponent of 2), undoing what a square does.When you encounter an exponent of 0.5, it indicates a square root. This is key to simplifying expressions like \(-81^{0.5}\), which transitions to \(-\sqrt{81}\). Understanding square roots not only aids in solving mathematical problems but also helps in various scientific calculations, where the notion of reversing a square often appears.Keep in mind that square roots typically yield positive results, but they can be troublesome when involving negative numbers, as seen in this expression.

Handling Negative Numbers

Negative numbers are numbers that are less than zero, represented with a minus sign (-) initially. They are crucial in expressing values that might be deficits or simply below zero, such as depths below sea level or temperatures below freezing.In mathematical expressions, the negative sign modifies the value it precedes. For example, in \(-\sqrt{81}\), the negative sign means you take the positive square root of 81, which is 9, and then multiply by -1, yielding -9. It is crucial to understand how to apply the negative sign correctly in operations.When dealing with negative products or roots, remember:

• A negative times a positive yields a negative.
• Two negatives multiplied together result in a positive.

Understanding these elementary rules about negative numbers can clarify and simplify seemingly complex calculations, making it easier to handle expressions involving both negative and positive values.