Problem 70 - Algebra 2

Question

Simplify each expression. \(81^{-0.25}\)

Step-by-Step Solution

Answer

The simplified form of \(81^{-0.25}\) is \(\frac{1}{3}\).

Step 1: Understand the effect of the negative exponent

A negative exponent indicates a reciprocal. Thus \(81^{-0.25}\) can be rewritten as \(\frac{1}{81^{0.25}}\).

Step 2: Understand the effect of the fractional exponent

A fractional exponent where the denominator is 2 represents a square root and where the denominator is 3 represents a cube root. In general, if the denominator of the fraction is \(n\), it represents the \(n\)th root. Here, the fractional exponent is 0.25 (which is \(\frac{1}{4}\)); this represents a fourth root. Thus \(\frac{1}{81^{0.25}}\) can be rewritten as \(\frac{1}{\sqrt[4]{81}}\).

Step 3: Calculate the fourth root of 81

The fourth root of 81 is 3. Therefore, \(\frac{1}{\sqrt[4]{81}}\) equals \(\frac{1}{3}\).

Key Concepts

Fractional Exponents

Fractional exponents might initially seem a bit tricky but they are actually very useful and simple once understood. Essentially, a fractional exponent contains a numerator and a denominator. The numerator is an ordinary power, and the denominator represents a root.
For example, if you see something like \[81^{0.25}\]The '0.25' can be expressed as \[\frac{1}{4}\].So, this tells us that instead of raising 81 to a power, we are taking the fourth root of 81.
Here’s how it works:

• Numerator = 1: This means the number itself, no extra multiplication.
• Denominator = 4: This indicates the fourth root.

Understanding fractional exponents makes both simplifying expressions and solving equations much easier. Remember that this applies whenever you see an exponent that's a fraction, so get comfortable with converting between exponentials and roots!

Fourth Root

The concept of the fourth root is symmetrically similar to finding square roots or cube roots, but specific to four. Essentially, if a number \( x \) is the fourth root of another number \( y \), then \( x^4 = y \).
When finding the fourth root of a number like 81:

• Consider what number times itself four times equals 81.
• Running through basic mental math, \( 3 \times 3 \times 3 \times 3 \) or \( 3^4 \) equals 81.

Hence, the fourth root of 81 is 3. By understanding this, you can find the fourth root quickly with practice and sometimes simple intuition. Keep practicing with different numbers until you feel comfortable switching between powers and roots interchangeably.

Reciprocal of a Power

A negative exponent might appear daunting at first glance, but it simply denotes taking the reciprocal of the base raised to the corresponding positive exponent. Simply put, \( x^{-a} = \frac{1}{x^a} \).This rule helps in simplifying exponentially involved problems.
In the expression \( 81^{-0.25} \), the negative exponent \(-0.25\)tells us that we need to take the reciprocal first:

• Convert \(81^{-0.25}\) to \(\frac{1}{81^{0.25}}\).

Once the reciprocal operation is done, you can proceed by working with the positive fractional exponent that remains, leading to easier calculations and simpler expressions. Remember, the negative exponent is nothing to fear; it’s all about flipping and then solving!