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Chapter 7: Problem 108

Simplify. Assume that \(x \geq 0 .\) \(\sqrt[12]{x^{44}}\)

Short Answer

Expert verified
Simplified form is \(\root{x^{\frac{11}{3}}}\).

Step by step solution

01

Express the Radical in Exponential Form

Rewrite the given radical expression using exponents. The general form is \(\root{n}\root{x^m} = x^{\frac{m}{n}}\). For \(\root{12}\root{x^{44}}\), it becomes \(\root{12}\root{x^{44}} = x^{\frac{44}{12}}\).
02

Simplify the Exponent Fraction

Simplify the fraction \(\frac{44}{12}\). The greatest common divisor (GCD) of 44 and 12 is 4. Divide both the numerator and the denominator by 4: \(\frac{44 \div 4}{12 \div 4} = \frac{11}{3}\).
03

Write the Simplified Expression

Now, express the simplified exponent: \(\root{12}\root{x^{44}}\) becomes \(\root{x^{\frac{11}{3}}}\).

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Key Concepts

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

radical expressions
Radical expressions involve roots, such as square roots, cube roots, and so on. In a radical expression, you might often see a symbol like \(\root{}\). For instance, the cube root of a number y would be written as \(\root[3]\root{y}\). Radical expressions can often be simplified using properties of exponents and roots. In the example \(\root[12]\root{x^{44}}\), the radical is a twelfth root. This means we are looking for a number that, when raised to the power of 12, equals \(x^{44}\).
exponential notation
Exponential notation expresses numbers as a base raised to an exponent. For example, in \(x^4\), 'x' is the base, and '4' is the exponent, meaning x multiplied by itself four times. In the given exercise, we used exponential notation to convert the radical expression into a more manageable form. We rewrite \(\root[12]\root{x^{44}}\) as \(x^{\frac{44}{12}}\). This conversion allows us to apply exponent rules and simplifies our calculations.
greatest common divisor (GCD)
The greatest common divisor (GCD) of two numbers is the largest number that divides both of them evenly. To simplify the exponent fraction \(\frac{44}{12}\), we find the GCD of 44 and 12. The GCD of 44 and 12 is 4. By dividing both the numerator and the denominator by their GCD, we simplify the fraction: \( \frac{44}{12} = \frac{44 梅 4}{12 梅 4} = \frac{11}{3}\). This gives us a simplified exponent for our expression, turning \(\root[12]\root{x^{44}}\) into \(x^{\frac{11}{3}}\).

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