Exponents represent an important concept in math that comes up frequently starting in algebra and continuing through higher level math and science courses. Exponents indicate repeated multiplication of a base number by itself a certain number of times. In this post, we will evaluate the exponential expression 2 to the negative 4th power, written as 2^-4. We will break down step-by-step how to calculate exponents with negative powers and discuss the key rules that govern these types of problems. Developing fluency with exponents lays the foundation for comprehending more complex exponential and logarithmic functions.

## What is 2 to the negative 4th power?

To evaluate 2 to the negative 4th power:

2^-4 means 1 divided by 2^4 2^4 = 16 Therefore, 2^-4 = 1/16

Here is a table showing the step-by-step work to evaluate 2 to the negative 4th power:

Step | Explanation |
---|---|

1 | 2^-4 means 2 to the negative 4th power |

2 | For negative exponents, a^-n = 1/a^n |

3 | So 2^-4 = 1 / 2^4 |

4 | Calculate 2^4 = 2 x 2 x 2 x 2 = 16 |

5 | Substitute 16 for 2^4: |

6 | 2^-4 = 1/16 |

To summarize:

- The general format is base^exponent
- Negative exponents mean to take the reciprocal of the positive exponent form
- Follow order of operations and rules for exponents
- 2 to the negative 4th power equals 1/16

## Understanding Exponents

Let’s first review some basics about exponents. The general structure of exponential expressions is:

Base number^Exponent

For example:

2^3 means 2 x 2 x 2 = 8

3^5 means 3 x 3 x 3 x 3 x 3 = 243

The exponent tells us how many times to multiply the base number by itself. This compact notation is useful for representing repeated multiplication neatly.

## Rules for Negative Exponents

When the exponent is negative, we apply these rules:

- a^-n means 1 divided by a^n
- a^0 always equals 1 for any base a

Keeping these rules in mind, let’s evaluate 2 raised to the -4 power.

Evaluating 2^-4

Following the rule for negative exponents:

2^-4 means 1 divided by 2^4

2^4 = 2 x 2 x 2 x 2 = 16

Therefore, 2^-4 = 1/16

We interpreted the negative exponent to mean flipping the fraction and putting 16 in the denominator.

## Checking Our Work

We can check that 1/16 is the correct evaluation of 2^-4 by entering it into a calculator:

2^(-4) = 0.0625

Which simplifies to 1/16, confirming our work.

## Key Takeaways for Exponents

Let’s summarize some key learnings:

- The exponent indicates the number of times to multiply the base number by itself.
- Negative exponents mean flipping the positive exponent fraction.
- Keep exponent rules handy until they become second nature.
- Take time to manually evaluate small exponential expressions.
- Check your work, especially when learning new exponent rules.
- Exponents underlie exponents, series, logarithms, calculus and more advanced math.

## Conclusion

In this post, we walked through evaluating the exponential expression 2^-4 step-by-step and discussed the reasoning behind treating negative exponents as flipped fractions. Developing conceptual understanding and fluency with basic exponential rules provides the foundation for higher math and science applications.

Consistently practicing calculating exponents strengthens skills needed for engineering, programming, finance, and other complex fields relying on exponential functions and power rules. Each solved exponential expression brings learners one step closer to mastery of mathematical notation and manipulation. Keep persevering through those exponents!

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