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Chapter 7: Problem 36

Evaluate each exponential. $$ 81^{-3 / 2} $$

Short Answer

Expert verified
\[ 81^{-3 / 2} = \frac{1}{729} \]

Step by step solution

01

- Understand the negative exponent

A negative exponent indicates that we take the reciprocal of the base raised to the absolute value of the exponent. So, we can rewrite the expression as \[ 81^{-3/2} = \frac{1}{81^{3/2}} \]
02

- Simplify the fractional exponent

The fractional exponent \(3/2\) can be split into two parts: the square root (denoted by 1/2) and the cube (denoted by 3). Thus: \[ 81^{3/2} = (81^{1/2})^3 \]
03

- Find the square root of 81

The square root of 81 is 9 because \(9^2=81\). So we get: \[ 81^{1/2} = 9 \]
04

- Raise the result to the power of 3

Now, we raise 9 to the power of 3: \[ 9^3 = 729 \]
05

- Take the reciprocal of 729

Since we originally had a negative exponent, we take the reciprocal of 729: \[ \frac{1}{729} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Negative Exponent
A negative exponent signifies that you need to find the reciprocal of the base raised to the absolute value of the exponent. For the expression \(81^{-3/2}\), this means you first deal with the negative sign by converting it as follows:

\[81^{-3/2} = \frac{1}{81^{3/2}}\]

This conversion helps in simplifying the expression since positive exponents are often easier to work with.
Fractional Exponent
Fractional exponents represent both root and power operations in one single term. Taking the fraction \(3/2\) from the example \(81^{3/2}\), you split it into two parts: the root (1/2) and the power (3).

Here's how to break it down:
  • First, find the square root of the base: \(81^{1/2}\).
  • Then, raise the result to the specified power: \((81^{1/2})^3\).


For the solution of the given example:
  • The square root of 81 is 9 as \(9^2 = 81\).
  • Then, you raise this result (9) to the power of 3, resulting in \(9^3 = 729\).
Exponentiation Steps
To solve exponential expressions step by step, follow these instructions:

Start by analyzing any negative exponents, converting them to a reciprocal expression. This makes working with positive exponents easier. Next, deal with any fractional exponents by breaking them into simpler operations of roots and powers. Finally, perform each operation in the given sequence to simplify the expression.

For our example:
  • Convert the negative exponent: \(81^{-3/2} = \frac{1}{81^{3/2}}\).
  • Simplify the fractional exponent: \(81^{3/2} = (81^{1/2})^3\).
  • Find the square root: \(81^{1/2} = 9\).
  • Raise to the power of 3: \(9^3 = 729\).
  • Reciprocate for the original negative exponent: \(\frac{1}{729}\).

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