Problem 541 - Elementary Algebra

Question

In the following exercises, simplify. (a) \((-243)^{\frac{1}{5}}\) (b) \(-243^{\frac{1}{5}}\) (c) (243) \(^{-\frac{1}{5}}\)

Step-by-Step Solution

Answer

(a) -3, (b) -3, (c) \frac{1}{3}

Step 1: Understand the difference in exponents

Identify that each part of the problem deals with exponents but has different signs and placements that significantly affect the simplification. Part (a) is \((-243)^{\frac{1}{5}}\), part (b) is \(-243^{\frac{1}{5}}\), and part (c) is \(243^{-\frac{1}{5}}\).

Step 2: Simplify Part (a)

For \((-243)^{\frac{1}{5}}\): We are finding the fifth root of \-243\. Since \-243\ is negative, we need to determine if there is a real number that, when raised to the power of 5, equals \-243\. The fifth root of a negative number is also negative, thus we get: \((-243)^{\frac{1}{5}} = -3\).

Step 3: Simplify Part (b)

For \(-243^{\frac{1}{5}}\): This means we first compute \(243^{\frac{1}{5}}\) and then apply the negative sign. The fifth root of \243\ is \3\, therefore the expression resolves to: \(-243^{\frac{1}{5}} = -3\).

Step 4: Simplify Part (c)

For \(243^{-\frac{1}{5}}\): Recall that a negative exponent means taking the reciprocal. Thus, we first find \(243^{\frac{1}{5}} \) which is \3\, and then take the reciprocal: \((243)^{-\frac{1}{5}} = \frac{1}{3}\).

Key Concepts

Negative Exponents

Understanding negative exponents is crucial for solving exercises like the provided example.
When we see a negative exponent, it means we have to take the reciprocal of the base raised to the positive exponent.
For instance, in part (c), the expression is \(243^{-\frac{1}{5}}\). A negative exponent transforms this into: \(\frac{1}{243^{\frac{1}{5}}}\).
Then we simplify the positive exponent, which gives us the fifth root of 243, resulting in 3. Therefore, \(243^{-\frac{1}{5}} = \frac{1}{3}\).
This principle can be extended to any negative exponent: \(a^{-n} = \frac{1}{a^n}\). Always remember, first remove the negative sign by converting to the reciprocal, then simplify the exponent.

Fifth Roots

Fifth roots can seem tricky at first, but they follow a straightforward principle.
Finding the fifth root of a number means identifying a value that, when raised to the power of 5, equals the original number.
For example, let's look at part (a) \((-243)^{\frac{1}{5}}\).
Here, we are looking for a number that when raised to the fifth power gives -243.
Negative numbers have real fifth roots since an odd number of negative factors results in a negative product.
In this case, we find that \(-3^5 = -243\), so \((-243)^{\frac{1}{5}} = -3 \).
Similarly, for part (b) \(-243^{\frac{1}{5}}\), we first compute the positive fifth root of 243, yielding 3, and then apply the negative sign, resulting in \(-3\).

Reciprocal of Exponents

The concept of reciprocal in exponents is closely related to negative exponents.
Simply put, a reciprocal flips the position of the number in a fraction.
For example, the reciprocal of 2 is \(\frac{1}{2}\).
In terms of exponents, this translates into transforming a negative exponent into a positive one by flipping the base.
For part (c) \(243^{-\frac{1}{5}}\), you are effectively finding the reciprocal of \(243^{\frac{1}{5}}\).
This means that \(243^{-\frac{1}{5}} = \frac{1}{243^{\frac{1}{5}}}\), and since \(243^{\frac{1}{5}} = 3\), we get \(243^{-\frac{1}{5}} = \frac{1}{3}\).
To sum up, the reciprocal of an exponent just requires taking the positive root and positioning it as a fraction.