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Chapter 10: Problem 138

\(\sqrt[4]{k^{20}}\)

Short Answer

Expert verified
\(\root{4}{k^{20}} = k^{5}\)

Step by step solution

01

Understand the problem

We need to simplify the expression \(\root{4}{k^{20}}\). This means we are looking for the fourth root of \(k^{20}\).
02

Use the root and exponent rule

Recall the rule that \( \root{n}{x^m} = x^{m/n} \). Applying this rule to our problem, we get: \(\root{4}{k^{20}} = k^{20/4}\).
03

Simplify the exponent

Simplify the exponent fraction \( 20/4 \). This gives us \(\frac{20}{4} = 5\). So, \(\root{4}{k^{20}} = k^{5}\).

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Key Concepts

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

Fourth Root
The fourth root of a number is a value that, when raised to the power of four, gives the original number. Just like the square root undoes squaring a number, the fourth root undoes raising a number to the fourth power. For instance, if you have \(\root{4}{16}\), it means you are looking for a number that, when multiplied by itself four times (\( x^4 \)), equals 16. In this case, that number is 2, because \((2^4 = 16)\). The fourth root is symbolized by a radical sign with an index of 4: \(\root{4}{ }\).
Exponent Rules
Exponent rules are fundamental in simplifying expressions involving powers. One of the key rules to remember is that \((a^m)^n = a^{m*n}\), meaning raising a power to another power multiplies the exponents. This rule is useful in breaking down or combining terms involving exponents. Another important rule is that \(\root{n}{a^m}\) can be rewritten using fractional exponents as \((a^{m/n})\). For example, in our exercise, \(\root{4}{k^{20}}\) was rewritten as \((k^{20/4})\).
Simplification of Expressions
Simplifying expressions means reducing them to their simplest form. This often involves combining like terms or reducing fractions. In the context of our exercise, the expression \(\root{4}{k^{20}}\) was simplified by converting the radical form to a fractional exponent and then dividing the exponent, as shown: \(\root{4}{k^{20}} = k^{20/4}\). Simplifying the fractional exponent \(20/4\) gives \((k^5)\), which is the simplest form of the original expression.
Fractional Exponents
Fractional exponents provide a way to rewrite roots as exponents. A fractional exponent \(a^{m/n}\) means taking the n-th root of \(a^m\). This concept is essential for converting roots into a format that might be easier to work with algebraically. For example, \(\root{4}{k^{20}}\) becomes \((k^{20/4})\) because the index 4 of the root becomes the denominator of the fractional exponent. Simplifying \(20/4\) results in 5, making our final answer \((k^5)\). Fractional exponents can make manipulating and simplifying expressions more straightforward.

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Most popular questions from this chapter

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