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Chapter 9: Problem 35

Find the indicated principal \(n\)th root. $$\sqrt[4]{10,000}$$

Short Answer

Expert verified
The principal fourth root of 10,000 is 10.

Step by step solution

01

Expressing in exponential form

Express the problem in a form that is easier to calculate. This can be done using the principle of the \(n\)th root where the \(n\)th root of a number \(x\) can be written as \(x^{\frac{1}{n}}\). Therefore, the 4th root of 10000 will be written as \(10000^{0.25}\).
02

Solving the exponent

Now that we have expressed the problem in a simpler form, we can solve the exponent. The key here is to remember that 'common' numbers often used in mathematics usually have perfect roots. For example, 10000 is a perfect square of 100 which in turn is a perfect square of 10, making 10 a perfect 4th root of 10000. Hence, \(10000^{0.25} = 10\).

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Key Concepts

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

Exponents
Exponents are a crucial concept in mathematics. They signify how many times a number, known as the base, is multiplied by itself. An exponent is displayed as a small number to the upper right of the base. For example, in the expression \(x^n\), \(x\) is the base and \(n\) is the exponent, indicating \(x\) is multiplied by itself \(n\) times. This operation is a shorthand for repeated multiplication, making calculations more convenient.
  • When the exponent is 2, it is called "squared," and when it is 3, it is "cubed."
  • Any number raised to the power of 1 remains unchanged.
  • If the exponent is zero, the result is always 1, assuming the base is not zero.
Understanding exponents is essential when dealing with nth roots, as they can help simplify radical expressions through exponential notation.
Principal Root
The principal root refers to the positive root of a number. For the nth root, it's the non-negative solution of \(x^{\frac{1}{n}}\). When we talk about finding the nth root of a number, we usually mean finding the principal root.
The notation \(\sqrt[n]{x}\) denotes taking the nth root of \(x\). This is equivalent to raising \(x\) to the power of \(\frac{1}{n}\) in exponential form. Principal root focuses on simplifying calculations by considering only the positive outcome.
  • It helps avoid confusion in everyday calculations where both positive and negative roots are possible.
  • The principal square root of a positive number is always positive, even though the square root operation itself could theoretically yield a negative counterpart.
By using the principal root, calculations become more straightforward, particularly when applied to powers and roots in solving equations.
Perfect Roots
Perfect roots are numbers that yield integers when their roots are calculated. These numbers occur frequently in mathematics due to their neat, whole-number results.
A perfect root scenario happens when the nth root of a number results in an integer. For example, a perfect square is a number whose square root is an integer, like 4, 9, or 16.
  • The concept is similar for cube roots: 8 and 27 are considered perfect cubes since their cube roots are 2 and 3, respectively.
  • Understanding perfect roots allows easier identification of numbers that simplify to whole numbers when roots are computed.
  • This concept also helps in deducing factors and simplifying complex calculations.
Occurrences of perfect roots make mathematical operations such as root finding more manageable, fostering a deeper understanding of numerical relationships.

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Most popular questions from this chapter

Exploration Enter any positive real number in your calculator and repeatedly take the square root. What real number does the display appear to be approaching?

Writing Give two reasons why it is important to check solutions of a radical equation.

Use the equation for the velocity of a free-falling object, \(v=\sqrt{2 g h}\), where \(v\) is measured in feet per second, \(g=32\) feet per second squared, and \(h\) is the height (in feet). An object strikes the ground with a velocity of 45 feet per second. Estimate the height from which it was dropped.

Solve the equation. $$\sqrt{3 x+4}-2=3$$

Factor the polynomial completely. $$x^{3}-27$$
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