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Chapter 10: Problem 86

simplify each expression. Include absolute value bars where necessary. $$\sqrt[6]{(-6)^{6}}$$

Short Answer

Expert verified
The simplified expression is \(|6|\)

Step by step solution

01

Apply the exponent

First, apply the exponent to the base, \((-6)^6\), which equals \(46656\). The negative sign is eliminated due to the even exponent, resulting in a positive number.
02

Apply the root

Next, take the sixth root of the result from Step 1. The sixth root of \(46656\) is \(6\) because \(6^6 = 46656\). However, since the root is even, the result would be an absolute value of \(6\), represented as \(|6|\) in the final expression.

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Key Concepts

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

Absolute Value
Understanding absolute value is important when dealing with expressions, because it tells us how far a number is from zero, without caring about direction. Therefore, the absolute value of a number is always non-negative.
  • If we have a number like \(-10\), the absolute value is written as \(|-10|\) and is equal to \(+10\).
  • For \(+10\), the absolute value is simply \(|10| = 10\).
In expressions involving roots and exponents, especially those with even roots, we apply absolute value to ensure our results are non-negative. Consider the expression \(\sqrt[6]{(-6)^{6}}\). When it simplifies to \(6\), the result is expressed in absolute value form, written as \(|6|\). This ensures the result respects the property of absolute values, even though it's already positive.
Even Exponents
Even exponents can be tricky but are highly important in simplifying expressions, especially when dealing with negative bases. When a number is raised to an even power, the result is always non-negative. This occurs because multiplying an even amount of negative numbers results in a positive product.
  • Take \((-2)^2\): \(-2 \times -2 = +4\); the negative signs cancel each other out.
  • In \((-3)^{4}\), similarly, multiplying out gives \(+81\) instead of \(-81\).
The expression \(\sqrt[6]{(-6)^{6}}\) features the base \(-6\) raised to the sixth power, an even number. Hence, it simplifies to a positive \(46656\). So, even exponents help us eliminate worry about negative results when simplifying roots of expressions.
Root Operations
Root operations are essential in simplifying expressions that involve exponents. They are the opposite of applying exponents and help revert an expression back to its base form. The **root** symbol \(\sqrt[n]{x}\) means to find the number which, when raised to the power of \(n\), gives \(x\).
  • For instance, \(\sqrt[3]{27}\) asks for the number which cubed equals 27. The answer is 3 because \(3^3 = 27\).
  • Similarly, \(\sqrt[4]{81}\) is solved by finding 3, since \(3^4 = 81\).
Applying this to the expression \(\sqrt[6]{46656}\), the task is to find a number that raised to power six yields \(46656\). That number is 6, because \(6^6 = 46656\). Thus, root operations are crucial for simplifying complex expressions back to their most reduced form.

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Most popular questions from this chapter

What is an extraneous solution to a radical equation?

Rationalize the denominator: \(\frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)

Use a graphing utility to solve each radical equation. Graph each side of the equation in the given viewing rectangle. The equation's solution set is given by the \(x\) -coordinate(s) of the point (s) of intersection. Check by substitution. $$\begin{aligned} &\sqrt{x}+3=5\\\ &[-1,6,1] \text { by }[-1,6,1] \end{aligned}$$

Add: \(\frac{2}{x-2}+\frac{3}{x^{2}-4}\) (Section 7.4, Example 7)

Simplify: \((\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}})^{2}\)
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