Problem 106 \(-\sqrt[4]{256}\)... [FREE SOLUTION]

Chapter 10: Problem 106

\(-\sqrt[4]{256}\)

Short Answer

Expert verified

-4

Step by step solution

Understand the expression

The expression given is \(-\sqrt[4]{256}\). This means we need to find the fourth root of 256 and then take the negative of that value.

Calculate the fourth root of 256

To find the fourth root of 256, we are looking for a number which, when raised to the power of 4, equals 256. We can express this as \(x^4 = 256\). By trial or using knowledge of powers, we find that \(4^4 = 256\), so the fourth root of 256 is 4.

Apply the negative sign

After finding the fourth root, which is 4, we need to apply the negative sign as indicated in the original expression. 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.

Understanding Negative Numbers

Negative numbers can be a bit tricky at first, but they're simply numbers less than zero. They are represented with a minus sign ( - ). For example,

-1
-5
-100

When computing with negative numbers, remember:

Adding a negative number is the same as subtracting its absolute value.
Multiplying two negative numbers gives a positive result.
Multiplying a negative number by a positive number gives a negative result.

Taking the example from the exercise, once you find the fourth root of 256, which is 4, you need to apply the negative sign. This changes the result from 4 to -4.

Exploring Radicals

Radicals, or roots, are ways to express that a number should be multiplied by itself a certain number of times to reach another number. For example, the square root of 9 is 3 because 3 multiplied by itself (3*3) equals 9.
Similarly, a fourth root, as in our exercise, asks for a number that when multiplied by itself 4 times gives 256. Expressing this using LaTeX: Determining this can be done through knowledge of common powers or trial and error if the numbers aren鈥檛 immediately clear.

Understanding Powers of Numbers

Powers of numbers involve expressing a number multiplied by itself a certain number of times. For example, 2 raised to the power of 3 ( 2^3 ) is equivalent to 2 * 2 * 2, which results in 8.
In our exercise, to find the fourth root of 256, we need to find a number that raised to the power of 4 gives 256. From our powers knowledge, 4 when raised to the power 4 equals 256: 4^4 = 256.
Expressing fourth roots can be done using exponents, such as: (256^{1/4}) This means finding the fourth root of 256. By understanding how to work with powers, finding the correct roots becomes intuitive.

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91影视

\(-\sqrt[4]{256}\)

Short Answer

Step by step solution

Understand the expression

Calculate the fourth root of 256

Apply the negative sign

Key Concepts

Understanding Negative Numbers

Exploring Radicals

Understanding Powers of Numbers

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