Problem 68 Simplify \(-\sqrt[5]{2048}\)... [FREE SOLUTION]

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Chapter 10: Problem 68

Simplify \(-\sqrt[5]{2048}\)

Short Answer

Expert verified

The simplest form is \( -2^{\frac{11}{5}} \).

Step by step solution

Identify the Expression

The given expression is \( -\sqrt[5]{2048} \). Identify the radical and the number inside the radical.

Factorize the Number

Factorize 2048 to its prime factors. 2048 can be written as \( 2048 = 2^{11} \).

Rewrite the Radical

Rewrite the expression using the prime factorization: \( -\sqrt[5]{2^{11}} \).

Simplify the Radical

Use the property of radicals that \( \sqrt[n]{a^m} = a^{\frac{m}{n}} \). Therefore, \( \sqrt[5]{2^{11}} = 2^{\frac{11}{5}} \).

Apply the Negative Sign

Finally, apply the negative sign: \( -2^{\frac{11}{5}} \). This is the simplest form.

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Key Concepts

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

Prime Factorization

Prime factorization is the process of breaking down a number into its prime number factors. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For example, the prime numbers from 1 to 20 are 2, 3, 5, 7, 11, 13, 17, and 19.

In our problem, we needed to factorize 2048. By repeatedly dividing by 2 (since 2 is a prime factor), we find that 2048 can be expressed as: \ 2048 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \ or more compactly as \(2^{11}\). This prime factorization makes it easier to simplify the radical in our next steps.

Properties of Radicals

Radicals involve roots, such as square roots, cube roots, and in our exercise, the fifth root. There are some helpful properties of radicals that make simplifying expressions involving roots easier.

The key property we used in our problem is:\[ \ \ \sqrt[n]{a^m} = a^{\frac{m}{n}} \]

This property allows us to convert a radical expression into an exponent. For example, \(\sqrt[5]{2^{11}}\) can be written as \(2^{\frac{11}{5}}\). Using this property helped us in transforming and simplifying the given expression.

Exponent Rules

Exponents are a way of expressing repeated multiplication of the same number. There are several rules for working with exponents that come in handy.

Some of the important rules include:

Product of Powers Rule: \(a^m \times a^n = a^{m+n}\)
Power of a Power Rule: \((a^m)^n = a^{m \times n}\)
Power of a Product Rule: \((ab)^n = a^n \times b^n\)
Quotient of Powers Rule: \(\frac{a^m}{a^n} = a^{m-n}\)

In our problem, the exponent rule allowed us to simplify \(\sqrt[5]{2^{11}}\) by converting the root into a fractional exponent, resulting in \(2^{\frac{11}{5}}\). Applying these rules properly helped us find the simplest form of the given expression.

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91影视

Simplify \(-\sqrt[5]{2048}\)

Short Answer

Step by step solution

Identify the Expression

Factorize the Number

Rewrite the Radical

Simplify the Radical

Apply the Negative Sign

Key Concepts

Prime Factorization

Properties of Radicals

Exponent Rules

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