Problem 125 \(\sqrt[5]{(-9)^{5}}\)... [FREE SOLUTION]

Chapter 10: Problem 125

\(\sqrt[5]{(-9)^{5}}\)

Short Answer

Expert verified

-9

Step by step solution

Understand the Radical

The expression \(\sqrt[5]{(-9)^{5}}\) involves a fifth root. The fifth root of a number is another number that, when raised to the power of 5, gives the original number.

Simplify the Expression

Simplify the expression inside the radical. Since the exponent and the root are the same, \((-9)^{5}\) raised to the fifth root simplifies to the base, which is \-9\. This is because any number raised to a given power and then taken the root of that same degree will return the original base number.

Final Answer

So, \(\sqrt[5]{(-9)^{5}}\) simplifies directly to \-9\.

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Key Concepts

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

fifth root

The fifth root is one of the key concepts in radical expressions. Finding the fifth root of a number means determining another number that, when raised to the power of 5, results in the original number. For example, the fifth root of 32 is 2 because \(2^5 = 32\). To express this algebraically, we use the notation \( \sqrt[5]{a}\), where 'a' is the number under the radical.

exponents

Exponents are a fundamental concept in algebra, representing repeated multiplication. For instance, \(2^3 = 2 \times 2 \times 2\), which equals 8. In the context of the problem \( \sqrt[5]{(-9)^{5}}\), the exponent is 5. The equation tells us that (-9) is multiplied by itself 5 times. Understanding exponents helps in simplifying expressions, especially when combining them with roots.

radical simplification

Radical simplification involves reducing a radical expression to its simplest form. In the given problem, \( \sqrt[5]{(-9)^{5}}\), we can simplify by recognizing that the exponent (5) and the root (5th root) are the same. This allows us to conclude that the fifth root simply cancels the fifth power, leaving us with the base, which is -9. Therefore, \ (-9)^5 \ and \sqrt[5]{(-9)^{5}\ directly simplify to -9.

algebra basics

Algebra basics form the foundation for solving more complex problems involving radicals and exponents. It includes understanding variables, constants, and the operations that can be performed on them. Key operations include addition, subtraction, multiplication, and division, as well as exponentiation and taking roots. A solid grasp of these basics enables you to manipulate and simplify expressions like \( \sqrt[5]{(-9)^{5}}\). Recognizing patterns and properties, such as \( (a^m)^n = a^{m \times n}\), is crucial for efficient problem-solving.

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91影视

\(\sqrt[5]{(-9)^{5}}\)

Short Answer

Step by step solution

Understand the Radical

Simplify the Expression

Final Answer

Key Concepts

fifth root

exponents

radical simplification

algebra basics

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