Problem 48 - Algebra A Combined Function

Question

Evaluate each exponential expression. $$ 81^{-3 / 4} $$

Step-by-Step Solution

Answer

The value of \(81^{-3/4}\) is \(\frac{1}{27}\).

Step 1: Write the Expression in Root Form

The given expression is \(81^{-3/4}\). First, express the negative exponent as a reciprocal and then rewrite it in root form. An expression with a fractional exponent, \(a^{m/n}\), can be rewritten as \(\sqrt[n]{a^m}\). So, \(81^{-3/4} = \frac{1}{81^{3/4}} = \frac{1}{(81^{1/4})^3}\).

Step 2: Find the Fourth Root of 81

Now, find the fourth root of 81, which is \(81^{1/4}\). Since \(81 = 3^4\), \(81^{1/4} = (3^4)^{1/4} \). Using the property \((a^m)^n = a^{m \cdot n}\), we get: \((3^4)^{1/4} = 3^{4 \cdot (1/4)} = 3^{1} = 3\).

Step 3: Cube the Result

Now that we have \(81^{1/4} = 3\), raise this result to the power of 3: \((81^{1/4})^3 = 3^3\). Calculate \(3^3\): \(3 \times 3 \times 3 = 27\).

Step 4: Take the Reciprocal

Finally, since \(81^{-3/4} = \frac{1}{(81^{1/4})^3}\), substitute the value found from the previous step: \(\frac{1}{27}\). This is the final solution to the expression.

Key Concepts

Fractional Exponents

Fractional exponents can initially seem confusing, but they unlock new ways to work with numbers. Essentially, these exponents act as a bridge between powers and roots. In mathematics, any number with a fractional exponent like \( a^{m/n} \) can be interpreted as taking the \( nth \) root of a raised to the \( mth \) power, written as \( \sqrt[n]{a^m} \). For instance, if you see \( 81^{3/4} \), it means taking the fourth root of 81 and then raising it to the power of 3. This efficient notation helps solve many kinds of problems more easily. It's a useful tool to simplify complex expressions into manageable parts.

Negative Exponents

Negative exponents might seem complex at first, but they're quite straightforward once you understand the concept. When you see a negative exponent like \( 81^{-3/4} \), it indicates taking the reciprocal of the base raised to the positive exponent. So, instead of calculating \( 81^{3/4} \), you find \( \frac{1}{81^{3/4}} \). It's a useful method for simplifying expressions without changing the underlying value, just transforming how it's represented. This operation helps rearrange and simplify equations, especially when fractions and decimals are involved.

Radicals

Radicals are symbols used to denote roots of a number, such as square roots, cubic roots, and more. The radical sign \( \sqrt{} \) represents the square root when there's no index, but it can signify higher-order roots with an index number. The expression \( \sqrt[n]{a} \) symbolizes the \( nth \) root of \( a \). In terms of fractional exponents, radicals and exponents correlate directly. For example, \( 81^{1/4} \) is equivalent to \( \sqrt[4]{81} \). Radicals are pivotal in simplifying equations, transforming complex exponent expressions into simpler radical forms.

Reciprocal Property

The reciprocal property is essential for understanding negative exponents. This property alters the base of an expression to its reciprocal value. Consider the expression \( a^{-n} \). By virtue of the reciprocal property, you convert this into \( \frac{1}{a^n} \). It allows the expression to remain equivalent while presenting it differently. This transformation is particularly helpful when simplifying expressions with complex exponent terms, as it eliminates negative signs in exponents and reframes the problem in a more solvable format. Through the reciprocal property, expressions become easier to handle and simplify.