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Chapter 5: Problem 68

Simplify each expression. Write answers using positive exponents. $$ \frac{3^{-4}}{1^{-9}} $$

Short Answer

Expert verified
The simplified expression is \( \frac{1}{81} \).

Step by step solution

01

Understand negative exponents

A negative exponent indicates reciprocal. For example, the expression \( a^{-b} \) can be written as \( \frac{1}{a^b} \). This means \( 3^{-4} \) is equivalent to \( \frac{1}{3^4} \) and \( 1^{-9} \) is equivalent to \( \frac{1}{1^9} \).
02

Simplify the denominator and numerator

The value \( 1^{-9} \) is simply \( 1 \) since any number raised to any power is 1. Therefore, the original expression \( \frac{3^{-4}}{1^{-9}} \) simplifies to \( \frac{1}{3^4} \) divided by \( 1 \).
03

Division of fractions

Dividing by 1 does not change a number's value, so \( \frac{1}{3^4} \div 1 \) simplifies to \( \frac{1}{3^4} \).
04

Evaluate the power of 3

Now compute \( 3^4 \), which is \( 3 \times 3 \times 3 \times 3 = 81 \).
05

Write the final expression

Substitute the evaluated power of 3 into the simplified expression to get \( \frac{1}{81} \).

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Key Concepts

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

Negative Exponents
The concept of negative exponents can initially be quite puzzling. However, understanding them is critical to simplifying expressions. A negative exponent essentially means that you take the reciprocal of the base and switch the exponent to positive. To illustrate this, think about the expression \( a^{-b} \). This is equivalent to \( \frac{1}{a^b} \).
  • \( 3^{-4} \) becomes \( \frac{1}{3^4} \)
  • \( 1^{-9} \) transforms into \( \frac{1}{1^9} \)
Understanding this conversion makes evaluating expressions with negative exponents much simpler, as you primarily deal with positive exponents, which are more straightforward to handle.
Reciprocal
When you encounter a reciprocal, it involves flipping a fraction upside down. In the realm of exponents, a negative exponent suggests that the base's reciprocal should be applied. In the example \( 3^{-4} \), you find its reciprocal which yields \( \frac{1}{3^4} \). For \( 1^{-9} \), once the flipping occurs, it's simply \( 1 \) because \( \frac{1}{1} = 1 \). Hence, the operation of taking reciprocals helps revert negative exponents to a format that is easier to evaluate.
  • Always flip to make the exponent positive.
  • Reciprocal helps in simplifying expressions.
Power of a Number
The power of a number refers to how many times you multiply that number by itself. For example, with the expression \( 3^4 \), it suggests multiplying 3 by itself four times: \[ 3 \times 3 \times 3 \times 3 = 81 \]Understanding this helps when dealing with exponents since you're essentially performing repeated multiplication. In the scenario where we simplified \( \frac{3^{-4}}{1^{-9}} \), knowing the power of 3 allowed us to write the final solution as \( \frac{1}{81} \).
  • \( 3^4 \) equals 81, achieved by multiplying 3 four times.
  • Exponents simplify complex expressions through repeated multiplication.

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