When you come across a fraction with a negative exponent, it can look a bit intimidating at first. But don’t worry—it’s actually quite simple once you understand the basic rules.
Understanding Negative Exponents
A negative exponent indicates that you need to take the reciprocal of the base. The reciprocal of a number is simply 1 divided by that number. For example, the reciprocal of 2 is $frac{1}{2}$, and the reciprocal of $frac{3}{4}$ is $frac{4}{3}$
General Rule
The general rule for any base $a$ raised to a negative exponent $-n$ is:
$a^{-n} = frac{1}{a^n}$
This means that you flip the base to its reciprocal and change the exponent to positive.
Applying the Rule to Fractions
When you have a fraction raised to a negative exponent, you apply the same rule. Let’s break it down with an example.
Example: $left(frac{2}{3}right)^{-2}$
- Reciprocal: First, take the reciprocal of the fraction $frac{2}{3}$, which is $frac{3}{2}$
- Positive Exponent: Change the negative exponent to a positive exponent.
So,
$left(frac{2}{3}right)^{-2} = left(frac{3}{2}right)^2$
- Solve: Now, raise the reciprocal to the positive exponent.
$left(frac{3}{2}right)^2 = frac{3^2}{2^2} = frac{9}{4}$
Therefore, $left(frac{2}{3}right)^{-2} = frac{9}{4}$
Another Example: $left(frac{5}{7}right)^{-3}$
- Reciprocal: Take the reciprocal of $frac{5}{7}$, which is $frac{7}{5}$
- Positive Exponent: Change the exponent to positive.
So,
$left(frac{5}{7}right)^{-3} = left(frac{7}{5}right)^3$
- Solve: Raise the reciprocal to the positive exponent.
$left(frac{7}{5}right)^3 = frac{7^3}{5^3} = frac{343}{125}$
Therefore, $left(frac{5}{7}right)^{-3} = frac{343}{125}$
Why Does This Work?
The concept of negative exponents is rooted in the properties of exponents and the rules of arithmetic. Essentially, a negative exponent tells you to perform the inverse operation of multiplication, which is division. By taking the reciprocal and changing the exponent to positive, you are effectively performing this inverse operation.
Practical Applications
Understanding how to work with negative exponents is crucial in various fields such as physics, engineering, and computer science. For instance, in physics, you might encounter formulas where you need to manipulate exponents to solve for a particular variable.
Conclusion
In summary, expressing a fraction with a negative exponent involves taking the reciprocal of the fraction and changing the exponent to positive. This simple rule can help you navigate through more complex mathematical problems with ease.