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Chapter 7: Problem 28

Find each root. $$ -\sqrt[4]{256} $$

Short Answer

Expert verified
-4

Step by step solution

01

Rewrite the Root Expression

Recognize that \(-\sqrt[4]{256}\) can be rewritten using exponential notation. Specifically, \(256\) as a power of 2: \(-\sqrt[4]{2^8}\).
02

Simplify the Radicand

Understand that \(\frac{8}{4} = 2\), so \(\sqrt[4]{2^8} = 2^2 = 4\). Therefore, \(-\sqrt[4]{256} = -4\).

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Key Concepts

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

Roots
Roots are an essential part of algebra that help you determine what number, when multiplied by itself a certain number of times, gives you the original number. In the exercise, we dealt with a fourth root. This means you are looking for a number that, when raised to the power of 4, results in 256.
In notation: \[-\sqrt[4]{256}\].
The fourth root is finding what value times itself four times equals 256. For example: \[(number)^4 = 256\].
This is simplified using exponents, and then your understanding of roots enables you to find the value.
Exponents
Exponents are a way to denote repeated multiplication of a number by itself. For instance, \[2^3\] means multiplying 2 by itself three times: \[2 \times 2 \times 2 = 8\].
In our problem, we rewrote 256 using exponents. We expressed it as \[2^8\] because 2 raised to the 8th power equals 256: \[2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 256\].
This step is crucial as it simplifies the process of taking the root and understanding the relationship between the base number and the exponent applied.
Radicals
Radicals are symbols like the square root (\[\sqrt{}\]), cube root (\[\sqrt[3]{}\]), and in this case, the fourth root (\[\sqrt[4]{}\]). Similarly, the radicals can be rewritten in exponential form to make them easier to work with.
In our example, we took the fourth root of 256. When you have \[-\sqrt[4]{2^8}\], you can simplify it by converting the radical into an exponent \[2^{8/4}\].
This converts the radical into a simple exponent which can then be worked with more easily. Finally, simplifying \[2^{8/4}\] results in \[2^2 = 4\], and don't forget the negative sign, leading to \[-4\].
Simplification
Simplification is the process of making an expression easier to understand or solve. It often involves reducing equations or expressions to their most basic elements.
This may involve combining like terms, reducing fractions, or, as in our problem, working through exponents and radicals.
In the given exercise, we simplified \[-\sqrt[4]{256}\]. By understanding that \[2^8\] can be decomposed to \[2^{8/4}\], we simplified the radical to directly getting \[2^2 = 4\], and kept the negative sign to arrive at \[-4\].
These steps break down the process to make the overall problem much easier to handle and understand.

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