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Chapter 10: Problem 108

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

Short Answer

Expert verified
The simplified form is \(x^{11/3}\).

Step by step solution

01

Recognize the Root and Exponent

Identify the expression as a radical with an index. The expression is \(\root(12){x^{44}}\). This is a 12th root of \(x^{44}\).
02

Convert to Exponential Form

Rewrite the radical using the exponent rule. \(\root(n){x^m} = x^{m/n}\). Therefore, \(\root(12){x^{44}} = x^{44/12}\).
03

Simplify the Fraction

Simplify \(\frac{44}{12}\). Find the greatest common divisor (GCD) of 44 and 12, which is 4. Divide both the numerator and the denominator by 4 to get \(\frac{44 \div 4}{12 \div 4} = \frac{11}{3}\).
04

Express in Simplified Form

Now, rewrite the expression with the simplified fraction: \(x^{11/3}\).

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Key Concepts

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

Radical Expressions
A radical expression involves roots, like square roots or cube roots. Think of it as the inverse of raising a number to a power. If you see a radical sign (鈭�), it means you're looking at a root. For example, \(\root(12){x^{44}}\) is a radical expression with an index of 12 and a radicand of \(x^{44}\). The index signifies which root you're taking, and the radicand is the value inside the radical. Radicals can be rewritten using exponents, which makes them easier to work with in many algebraic operations.
Exponent Rules
Exponents are a way to show repeated multiplication. For instance, \(x^3\) means \(x \times x \times x\). When dealing with radicals, you can often use exponent rules to make the expressions easier to handle. One useful rule is converting a radical to an exponent form: \[ \root(n)(x^m) = x^{\frac{m}{n}} \]. This means that instead of dealing with the 12th root of \(x^{44}\), you can rewrite it as \(x^{44/12}\). Using exponent rules helps simplify complex expressions and make calculations more straightforward.
Fraction Simplification
Simplifying fractions means reducing them to their simplest form. To do this, divide the numerator and the denominator by their greatest common divisor (GCD). In the example \( \frac{44}{12}\), both numbers are divisible by 4, which is their GCD. So, \( \frac{44 \div 4}{12 \div 4} = \frac{11}{3} \), giving you a simpler fraction. Simplifying fractions is crucial for making algebraic expressions easier to understand and work with.
Greatest Common Divisor
The greatest common divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder. To simplify \( \frac{44}{12} \), find their GCD. The factors of 44 are \{1, 2, 4, 11, 22, 44}\, and the factors of 12 are \{1, 2, 3, 4, 6, 12}\. The largest number in both sets is 4. Dividing both 44 and 12 by their GCD of 4 simplifies the fraction to \( \frac{11}{3} \). Mastering the GCD helps in many areas of math, particularly in simplifying fractions.

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Most popular questions from this chapter

The windchill factor is a measure of the cooling effect that the wind has on a person's skin. It calculates the equivalent cooling temperature if there were no wind. The National Weather Service uses the formula $$ \text { Windchill temperature }=35.74+0.6215 T-35.75 V^{4 / 25}+0.4275 T V^{4 / 25} $$ where T is the temperature in \(^{\circ} \mathrm{F}\) and \(\mathrm{Vi}\) is the wind speed in miles per hour, to calculate wind- chill. The table gives the windchill factor for various wind speeds and temperatures at which frostbite is a risk, and how quickly it may occur. Use the formula to determine the windchill temperature to the nearest tenth of a degree, given the following conditions. Compare answers with the appropriate entries in the table. \(30^{\circ} \mathrm{F}, 15\) -mph wind

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A Panasonic Smart Viera E50 LCD HDTV has a rectangular screen with a 36.5 -in. width. Its height is 20.8 in. What is the length of the diagonal of the screen to the nearest tenth of an inch? (Data from measurements of the author's television.)

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