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Chapter 7: Problem 65

Simplify each root. $$ -\sqrt[6]{(-5)^{6}} $$

Short Answer

Expert verified
-5

Step by step solution

01

Understand the Root and Exponent Relationship

To simplify \(-\sqrt[6]{(-5)^{6}}\), recognize that taking the sixth root of \((-5)^{6}\) is equivalent to finding \((-5)^{6/6}\).
02

Simplify the Exponent

Simplify the exponent \(6/6\) to \(1\). Therefore, \((-5)^{6/6} = -5^{1} \).
03

Apply the Root

Applying the sixth root gives us \(-(-5) = -5\).

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Key Concepts

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

Exponentiation
Exponentiation is the operation of raising one number, the base, to the power of another number, the exponent. In this exercise, \( (-5)^6\), -5 is the base and 6 is the exponent. This means we multiply -5 by itself six times. Understanding how exponents work will help you handle more complex problems.
Here are a few key points to remember:
  • The exponent tells you how many times to multiply the base by itself.
  • Any number raised to an exponent of 1 is just the number itself. For example, \( 5^1 = 5 \).
  • A number raised to the power of 0 is always 1. For instance, \( 9^0 = 1 \).
Keep these rules in mind when working with exponents.
Roots
Roots are the inverse operation of exponentiation. Whereas exponentiation involves multiplying a number by itself a certain number of times, finding a root involves finding a number which, when multiplied by itself a certain number of times, gives you the original number.
In the problem, we deal with a sixth root. To simplify \( - \sqrt[6]{(-5)^6} \), realize we're looking for a number that, when raised to the power of 6, returns to \( -5 \).
  • The square root \( \sqrt{x} \) is the opposite of squaring a number.
  • Similarly, the cube root \( \sqrt[3]{x} \) is the opposite of cubing a number.
  • The sixth root \( \sqrt[6]{x} \) means we're searching for a number which, when raised to the sixth power, equals x.
Understanding roots helps in reversing operations done by exponents.
Simplification
Simplification involves reducing a mathematical expression to its simplest form. In this exercise, we simplified \( - \sqrt[6]{(-5)^6} \) step-by-step:
First, we recognized the relationship between exponents and roots. Taking the sixth root of \( (-5)^6 \) is the same as \( (-5)^{6/6} \). The fraction simplifies since \( 6/6 \) equals 1.
This results in \( (-5)^1 = -5 \). Finally, we apply the sixth root and get -5.
Here are some tips on simplification:
  • Always look for ways to reduce fractions in the exponent.
  • Root and power relationships can often simplify complex expressions quickly.
  • Practice breaking down steps clearly, as shown, to avoid confusion.
Simplification makes problems easier to solve and understand.

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Most popular questions from this chapter

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt{y} \cdot \sqrt[3]{y z} $$

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{2}{3 \sqrt{5}+2 \sqrt{3}} $$

Find the distance between each pair of points. \((\sqrt{2}, \sqrt{6})\) and \((-2 \sqrt{2}, 4 \sqrt{6})\)

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt{\sqrt[3]{\sqrt[4]{x}}} $$

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{3 \sqrt{x}}{\sqrt{x}-2 \sqrt{y}} $$
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