Problem 57 - Algebra 2

Question

Simplify each number. $$64^{-0.5}$$

Step-by-Step Solution

Answer

The simplified form of \(64^{-0.5}\) is \(\frac{1}{8}\).

Step 1: Understand the exponent's rule

Two rules can be used to simplify the given expression. First, any number raised to the power of -n is equivalent to 1 divided by that number raised to the power of n. The formula for this is \(a^{-n} = \frac{1}{a^n}\). Second, any number raised to the power of 0.5 equals the square root of that number. The formula for this is \(a^{0.5} = \sqrt{a}\).

Step 2: Apply the negative exponent rule

Apply the first rule: \(a^{-n} = \frac{1}{a^n}\). For our equation, \(64^{-0.5}\), this gives us \(\frac{1}{64^{0.5}}\).

Step 3: Apply the fractional exponent rule

Next, apply the second rule: \(a^{0.5} = \sqrt{a}\). This changes the expression to \(\frac{1}{\sqrt{64}}\).

Step 4: Calculate the square root

We find that the square root of 64 equals 8, so the expression simplifies to \(\frac{1}{8}\).

Key Concepts

Understanding Negative Exponents

Negative exponents can be a bit tricky, but let's break it down into simpler parts. A negative exponent means that you can take the reciprocal of the base and make the exponent positive. Essentially, it is the opposite of multiplying that number several times.
For instance, when you have a number like \(a^{-n}\), it becomes the reciprocal of that number raised to the positive power, \(\frac{1}{a^n}\).
This rule helps us manage large calculations or transforms negative powers into something more familiar. So in our exercise, \(64^{-0.5}\) converts into \(\frac{1}{64^{0.5}}\) thanks to our negative exponent rule, setting us up for further simplification.

Exploring Fractional Exponents

Fractional exponents might look intimidating, but they offer a convenient way to express roots. A fractional exponent is an exponent that is a fraction, where the numerator represents a power and the denominator indicates a root.
For example, in \(a^{0.5}\), the 0.5 is the fractional exponent and tells us to take the square root of \(a\). Essentially, \(a^{0.5}\) is the same as \(\sqrt{a}\).
This allows for simplified expressions. So in our problem, once we transform \(64^{-0.5}\) to \(\frac{1}{64^{0.5}}\), it further simplifies to \(\frac{1}{\sqrt{64}}\) using the fractional exponent rule.

Calculating the Square Root

The final step in our simplification is calculating the square root of a number. The square root is a special number that, when multiplied by itself, gives the original number.
Finding the square root is straightforward if you know your basic squares. For example, you know \(8 \times 8 = 64\), so the square root of 64 is 8.
In our exercise, this means transforming \(\frac{1}{\sqrt{64}}\) into \(\frac{1}{8}\). This gives us the final result, providing a number that makes sense and is easy to understand. The journey from negative exponents to fractional exponents and finally calculating the square root ensures accurate results and a clear understanding of exponent rules.