Problem 10 True or False \(\sqrt[4]{(-3)^{4... [FREE SOLUTION]

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Chapter 0: Problem 10

True or False \(\sqrt[4]{(-3)^{4}}=-3\)

Short Answer

Expert verified

False, the fourth root of 81 is 3, not -3.

Step by step solution

Understand the Problem

The problem asks to determine if the statement \(\sqrt[4]{(-3)^{4}}=-3\) is true or false.

Evaluate the Expression Inside the Root

Since \[ (-3)^4 = (-3) \times (-3) \times (-3) \times (-3) \], we need to calculate this value first. \[ (-3) \times (-3) = 9 \] \[ 9 \times (-3) = -27 \] \[ -27 \times (-3) = 81 \]. So, \((-3)^4 = 81\).

Compute the Fourth Root

Next, evaluate the fourth root of 81. \[ \sqrt[4]{81} = 3 \]

Compare the Result to the Given Statement

The statement given is \(\sqrt[4]{(-3)^{4}} = -3\). Since \(\sqrt[4]{81} = 3\), the statement says the value is negative, but it should be positive.

Conclusion

The statement \(\sqrt[4]{(-3)^{4}} = -3\) is false because the fourth root of 81 is 3, not -3.

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Key Concepts

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

Exponents

Exponents represent repeated multiplication.
When we say \(a^n\), 'a' is the base, and 'n' is the exponent. This means we multiply 'a' by itself 'n' times.
For example, for \((-3)^4\), it means \((-3) \times (-3) \times (-3) \times (-3)\).
Using exponents helps simplify large numbers and expressions.

In our problem, \((-3)^4 = 81\).
Notice how the negative base raised to an even power results in a positive outcome.

Radicals

Radicals are symbols used to represent roots of numbers. The most common radical is the square root, but there are higher-order roots, too.
The radical symbol (\(\sqrt{}\)) is used here.
In \(\sqrt[4]{81}\), the '4' indicates the fourth root, meaning \((a)^4 = 81\).

For \(81\), we need to find a number (a) such that \(a^4 = 81\).
The solution gives \(a = 3\), so \(\sqrt[4]{81} = 3\).

Remember, when dealing with even roots, we get a positive number as the result.

Real Numbers

Real numbers include all the numbers on the number line: positive, negative, fractions, and decimals.

In our problem, numbers \(-3\), \(3\), and \(81\) are all real numbers.

Real numbers do not include imaginary numbers.
Understanding real numbers helps in dealing with roots and powers, as seen in the example: \(\sqrt[4]{(-3)^4}\).

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91影视

True or False \(\sqrt[4]{(-3)^{4}}=-3\)

Short Answer

Step by step solution

Understand the Problem

Evaluate the Expression Inside the Root

Compute the Fourth Root

Compare the Result to the Given Statement

Conclusion

Key Concepts

Exponents

Radicals

Real Numbers

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