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Chapter 7: Problem 22

Write each radical using rational exponents. $$ \sqrt[3]{62} $$

Short Answer

Expert verified
The expression \( \sqrt[3]{62} \) is written as \( 62^{\frac{1}{3}} \) using rational exponents.

Step by step solution

01

Convert Radical to Fractional Exponent

The expression \( \sqrt[3]{62} \) represents the cube root of 62. According to the properties of exponents, a cube root can be expressed as raising a number to the power of \( \frac{1}{3} \). So, we rewrite the expression by using a fractional exponent: \( 62^{\frac{1}{3}} \).

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Key Concepts

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

Exponents
Exponents are a fundamental concept in mathematics representing repeated multiplication of a number by itself. For instance, when we say \( a^3 \), it means the number \( a \) is multiplied by itself three times. Exponents are not limited to whole numbers; they can also be fractions. This concept allows us to perform operations on very large or small numbers efficiently.
By using exponents, calculations become simpler:
  • \( a^1 \) is just \( a \) itself.
  • \( a^0 = 1 \), provided \( a \) is not zero.
  • Products of the same base are added, like \( a^m \times a^n = a^{m+n} \).
It鈥檚 essential to understand these basic rules, as they form the foundation for more complex operations like radicals and fractional exponents.
Radicals
Radicals involve roots, such as square roots or cube roots, and are the opposite of using exponents. The radical symbol \( \sqrt{} \) represents a root, with an index indicating which root.
For example:
  • The square root of \( x \) is \( \sqrt{x} \), where no index is usually written, implying a square root.
  • Cube roots, like \( \sqrt[3]{x} \), indicate the number that, when cubed, yields \( x \).
The transition to dealing with radicals more flexibly often involves converting them to exponents. Each radical expression can correspond to an expression with exponents.
Though initially appearing complex, recognizing how to switch between the two formats simplifies calculations significantly, making mathematics more manageable.
Fractional Exponents
Fractional exponents are a combination of exponents and radicals. They offer an alternative way to express root operations. For any radical expression \( \sqrt[n]{a} \), it can be written in terms of fractional exponents as \( a^{\frac{1}{n}} \).
This conversion is based on the definition of roots as fractional powers:
  • \( \sqrt{a} = a^{\frac{1}{2}} \)
  • \( \sqrt[3]{a} = a^{\frac{1}{3}} \)
  • In a more general form, \( \sqrt[n]{a} = a^{\frac{1}{n}} \)
Fractional exponents allow easier manipulation and combination of radicals in equations since they adhere to all the rules of exponents. Understanding how to convert radicals to fractional exponents is crucial for simplifying expressions and solving equations efficiently in algebra.

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Most popular questions from this chapter

For Exercises 69 and \(70,\) use the following information. There are three temperature scales: Fahrenheit \(\left(^{\circ} \mathrm{F}\right),\) Celsius \(\left(^{\circ} \mathrm{C}\right)\) and Kelvin (K). The function \(K(C)=C+273\) can be used to convert Celsius temperatures to Kelvin. The function \(C(F)=\frac{5}{9}(F-32)\) can be used to convert Fahrenheit temperatures to Celsius. Find the temperature in Kelvin for the boiling point of water and the freezing point of water boils at \(212^{\circ} \mathrm{F}\) and freezes at \(32^{\circ} \mathrm{F} .\)

Write each radical using rational exponents. $$ \sqrt[7]{5^{3}} $$

Simplify. $$ \sqrt{4 x^{3} y^{2}} $$

REASONING Determine whether the equation \(\frac{\sqrt{\left(x^{2}\right)}^{2}}{-x}=x\) is sometimes, always, or never true when \(x\) is a real number. Explain your reasoning.

Simplify. $$ \left(a^{2}+a+1\right)(a-1) $$
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