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Chapter 9: Problem 56

Evaluate each of the following. \(81^{-3 / 4}\)

Short Answer

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

Step by step solution

01

Understand the problem

We are asked to evaluate the expression \(81^{-3/4}\). This involves working with negative and fractional exponents.
02

Convert the base to a power of 3

Recognize that 81 is a perfect power that can be rewritten as such. Since \(81 = 3^4\), we can substitute this back into the original expression, giving us \((3^4)^{-3/4}\).
03

Apply the Power of a Power Rule

The rule for exponents \((a^m)^n = a^{m \cdot n}\) allows us to simplify \((3^4)^{-3/4}\) to \(3^{(4 \cdot -3/4)}\).
04

Simplify the exponent

Multiply the exponents: \(4 \cdot -3/4 = -3\). The expression then simplifies to \(3^{-3}\).
05

Evaluate the expression with a negative exponent

The rule for negative exponents \(a^{-n} = \frac{1}{a^n}\) allows us to rewrite \(3^{-3}\) as \(\frac{1}{3^3}\).
06

Calculate \(3^3\)

Calculate \(3^3\) which is \(3 \times 3 \times 3 = 27\).
07

Write the final result

The entire expression \(3^{-3}\) simplifies to \(\frac{1}{27}\), which is our final answer.

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Key Concepts

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

Power of a Power Rule
The power of a power rule is an essential tool when dealing with exponential expressions. It provides a simple way to simplify expressions where an exponent is raised to another exponent. When you see
  • an expression like \((a^m)^n\),
  • it can be simplified using the rule to \(a^{m \cdot n}\).
This means we multiply the exponents together. This rule is incredibly useful for simplifying complex expressions before evaluating them, making calculations much easier. For instance, in our original problem, converting \((3^4)^{-3/4}\) to \(3^{(4 \cdot -3/4)}\) streamlines the expression remarkably well.
Perfect Power
Recognition of perfect powers can simplify your calculations significantly. Perfect powers are expressions that can be written as another number raised to a whole number exponent. For example, 81 is a perfect power because it can be expressed as \(3^4\).
  • This insight allows you to rewrite expressions,
  • making the use of exponent rules even more straightforward.
In simplifying expressions such as \(81^{-3/4}\), recognizing 81 as a power of 3 is key. By rewriting the base as a perfect power, the entire expression becomes much easier to handle using subsequent rules like the power of a power rule.
Negative Exponent Rule
The negative exponent rule is a fundamental aspect of working with exponents. It allows us to transform expressions with negative exponents into more manageable forms. According to this rule:
  • the expression \(a^{-n}\) is the same as \(\frac{1}{a^n}\).
This rule helps in keeping expressions positive and less complicated. In our example, after applying the power of a power rule, we simplified \(3^{-3}\) to \(\frac{1}{3^3}\), making it easier to evaluate further. Understanding and applying the negative exponent rule helps to convert any negative exponents into their positive counterparts.
Evaluating Expressions
Evaluating expressions with exponents requires a step-by-step approach to simplify and compute accurately. Here鈥檚 how to handle such expressions effectively:
  • First, identify any opportunities to rewrite numbers as perfect powers, such as expressing 81 as \(3^4\).
  • Next, apply relevant rules like the power of a power rule to simplify the expression further.
  • Then, use the negative exponent rule to make sure all exponents are positive before final calculations.
  • Finally, compute the simplified expression by handling any remaining arithmetic, like \(3^3 = 27\).
In the given problem, careful rewriting, simplification, and evaluation helped in deriving the final expression as \(\frac{1}{27}\). Breaking down the process into manageable steps makes the evaluation of any expression with negative and fractional exponents clear and understandable.

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Most popular questions from this chapter

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=e^{x+4} $$

Find the value of each logarithmic expression. See Examples 3 and 5. $$ \log _{2} \frac{1}{32} $$

Solve. See Example 4. $$ \log _{2} 16=x $$

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