Problem 30 - Intermediate Algebra

Question

Write with positive exponents. Simplify if possible. $$ 64^{-2 / 3} $$

Step-by-Step Solution

Answer

\(\frac{1}{16}\)

Step 1: Rewrite using the Positive Exponent

The expression given is \(64^{-2/3}\). To rewrite this with a positive exponent, we use the reciprocal property of exponents: \(a^{-b} = \frac{1}{a^b}\). Therefore, \(64^{-2/3} = \frac{1}{64^{2/3}}\).

Step 2: Consider the Cube Root

Now, we need to simplify \(64^{2/3}\). The fractional exponent \(64^{2/3}\) means we first take the cube root of 64 and then square the result. Recall that \(64 = 4^3\). Therefore, \(64^{1/3} = 4\).

Step 3: Square the Result from the Root

Having found the cube root to be 4 in the previous step, the next task is to square 4. Thus, \(4^2 = 16\). This means \(64^{2/3} = 16\).

Step 4: Write the Simplified Expression

We substitute \(64^{2/3} = 16\) back into the expression \(\frac{1}{64^{2/3}}\) to get \(\frac{1}{16}\). This is the simplest form of the given expression.

Key Concepts

Positive Exponents

In math, dealing with positive exponents is all about positivity! When an exponent is positive, it suggests that we multiply the base with itself the number of times shown by the exponent. For example, with the expression \(a^b\), if \(b\) is a positive number, you'd multiply \(a\), \(b\) times.
This results in a larger number if the base \(a\) is greater than 1. Working with positive exponents makes our calculations straightforward.

• Expression: \(3^2\)
• Calculation: \(3 \times 3 = 9\)

Dealing with positive exponents is like stepping forward. Every step (or multiplication) takes you further from the start but in a predictable direction, unlike dipping into negatives or fractions.

Fractional Exponents

Fractional exponents offer a dynamic way to express roots and powers in a single step. Let's demystify this a little. A fractional exponent like \(a^{m/n}\) means you take the \(n\)-th root of \(a\) and then raise the result to the \(m\)-th power.
Consider \(64^{2/3}\):

• Start by finding the cube root, since our denominator is 3. \(64^{1/3} = 4\) because \(4^3 = 64\).
• Next, take that result, 4, and square it because our numerator is 2: \(4^2 = 16\).

Fractional exponents can seem tricky at first, but they streamline the process of working with roots and powers in tandem.

Simplifying Expressions

Simplifying expressions ensures that we are working with the most efficient or concise version of a number or equation. Simplification might involve turning a complex or cumbersome expression into something more digestible.
When simplifying \(64^{-2/3}\):

• We start by converting the negative exponent to its reciprocal form: \(\frac{1}{64^{2/3}}\).
• Then we simplify \(64^{2/3}\) by first finding the cube root of 64 and then squaring the result, leading us to \(\frac{1}{16}\).

Breaking it down step-by-step prevents errors and helps you see the beauty and symmetry often hidden within math operations.

Reciprocal Property of Exponents

The reciprocal property of exponents is a powerful tool when dealing with negative exponents. The property states that \(a^{-b} = \frac{1}{a^b}\). This means if you face a negative exponent, you can "flip" it into the denominator, turning it positive.
Using \(64^{-2/3}\) as an example, the reciprocal property came to the rescue:

• Transformation: \(64^{-2/3} = \frac{1}{64^{2/3}}\)
• This step is crucial because it transforms a negative exponent into a manageable positive one.

By using this property, simplifying becomes less daunting. Negative exponents might make expressions look complex, but they can be made simpler by just being flipped.