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Chapter 9: Problem 453

In the following exercises, simplify. (a) \(\sqrt[8]{k^{8}}\) (b) \(\sqrt[6]{p^{6}}\)

Short Answer

Expert verified
(a) \( k \) (b) \( p \)

Step by step solution

01

- Understand the Problem

Determine the roots of exponential expressions given in the problem. The goal is to simplify these expressions by removing the radicals.
02

- Simplify \( \sqrt[8]{k^{8}} \)

When raising a variable to the same power as the root, the radical and the exponent cancel each other out. Therefore, \( \sqrt[8]{k^{8}} = k \).
03

- Simplify \( \sqrt[6]{p^{6}} \)

Similarly, for \( \sqrt[6]{p^{6}}\), the sixth root and the sixth power cancel each other out: \( \sqrt[6]{p^{6}} = p \).
04

- Write the Final Simplified Expressions

Combine the simplified expressions from Steps 2 and 3: \( \sqrt[8]{k^{8}} = k \) and \( \sqrt[6]{p^{6}} = p \).

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Key Concepts

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

roots of exponential expressions
To solve these problems, you need to understand the roots of exponential expressions. When we take the root of an exponential expression, we are looking for a base number that, when raised to a certain power, equals the expression inside the radical. For example, the expression \(\root[3]{8}\) asks for a number which, when raised to the power of 3, gives 8. The answer here is 2, because \(2^3 = 8\). This logic extends to any root and exponent, helping simplify more complex expressions like the ones in our problem: \( \root[8]{k^{8}}\) and \( \root[6]{p^{6}}\).
simplification of radicals
Simplifying radicals involves understanding how roots and exponents interact. When the exponent inside the radical matches the root, they effectively cancel out. For instance, \( \root[8]{k^{8}} \) simplifies to just \k\. This is because taking the 8th root of \k^8\ gives us back the base number \k\. Think of it like this: the radical and the exponent undo each other. It's like performing and then reversing an action. It's a straightforward, one-step simplification. Therefore, when you see \(\root[6]{p^{6}} \), you know it simplifies directly to \p\.
cancelling exponents and radicals
An important technique in simplifying radical expressions is the concept of canceling exponents and radicals. When the radical's index (the small number outside the root symbol) matches the exponent on the variable inside, they neutralize each other. For instance: if you have an expression \( \root[n]{a^{n}}\), it will simplify to just \a\. This happens because the \-th root and the \-th power are inverse operations. They undo each other completely. Applying this to our problems, \( \root[8]{k^{8}} = k\) and \( \root[6]{p^{6}} = p\). This makes complex-looking expressions very simple to work with.

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Most popular questions from this chapter

In the following exercises, solve. \(\sqrt{4 x-3}=7\)

In the following exercises, simplify by rationalizing the denominator. $$ \frac{3}{5+\sqrt{5}} $$

(a) Approximate \(\frac{1}{\sqrt{2}}\) by dividing \(\frac{1}{1.414}\) using long division without a calculator. (b) Rationalizing the denominator of \(\frac{1}{\sqrt{2}}\) gives \(\frac{\sqrt{2}}{2}\). Approximate \(\frac{\sqrt{2}}{2}\) by dividing \(\frac{1.414}{2}\) using long division without a calculator. (C) Do you agree that rationalizing the denominator makes calculations easier? Why or why not?

In the following exercises, solve. $$ 2 \sqrt{5 x+1}-8=0 $$

In the following exercises, simplify by rationalizing the denominator. $$ \frac{\sqrt{5}}{\sqrt{y}-\sqrt{7}} $$
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