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Chapter 1: Problem 30

Convert to radical notation. \(t^{1 / 7}\)

Short Answer

Expert verified
\(t^{1/7} = \sqrt[7]{t}\)

Step by step solution

01

Identify the exponent

The given expression is in the form of a rational exponent: \(t^{1/7}\).
02

Rewrite the rational exponent

Convert the rational exponent to radical notation. The general form to convert is \(a^{1/n} = \sqrt[n]{a}\).
03

Apply the conversion

Apply this rule to the given expression: \(t^{1/7} = \sqrt[7]{t}\).

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Key Concepts

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

rational exponents
Rational exponents can seem complicated at first, but they are just another way to represent roots. Instead of using the root symbol (鈭�), we use fractions as exponents. For instance,
  • The exponent \( \frac{1}{2} \) means a square root.
  • The exponent \( \frac{1}{3} \) means a cube root.
  • The exponent \( \frac{1}{n} \) means an \( n \)th root.
Here, in the given problem, \( t^{\frac{1}{7}} \) uses a rational exponent to denote a seventh root. This can be more convenient than writing lengthy radicals, especially in more complex expressions.
conversion formulas
Conversion formulas help translate expressions from one form to another. For rational exponents and roots, a key formula is \( a^{\frac{1}{n}} = \sqrt[n]{a} \). This tells us that:
  • The denominator of the fraction becomes the index of the root.
  • The numerator indicates the power.
Using this, any rational exponent can be turned into a radical. For example, \( t^{\frac{1}{7}} \) is converted to radical form as \ \sqrt[7]{t} \.
mathematical notation
Mathematical notation is the language of math. Mastery of notation helps in understanding and solving problems efficiently. Let's break down the notation used here:
  • Exponents: Write numbers in the power form, like \( x^b \), where \ b \ is the exponent.
  • Radicals: Use the root symbol \( \sqrt[]{} \), with an optional index for roots other than square roots.
  • Fractions in exponents: Indicate both power and root (e.g., \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \)).
This standardized language allows us to consistently convert and solve mathematical expressions, like converting \( t^{\frac{1}{7}} \) to \ \sqrt[7]{t} \ in the original exercise.

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Most popular questions from this chapter

A key \(n\) notes (including half-notes). above the A above middle C generates a sound wave proportional to $$ y=\sin \left(880 \cdot 2^{n / 12} \pi t\right) $$ where \(t\) is in seconds. Use this equation to solve the following exercises. Find the note with a frequency of \(4 \cdot 440=1760 \mathrm{~Hz}\)

Use a calculator to find the radian measure of an acute angle whose trigonometric function is given. \(\tan t=1.26\)

Use a calculator to find the radian measure of an acute angle whose trigonometric function is given. \(\sin t=0.59\)

Oscillations with Trends. Graph the following functions on the interval \(0 \leq t \leq 10\). $$ f(t)=t^{2} / 10+3+4 \sin 5 t $$

Use a calculator to find the values of the following trigonometric functions. \(\csc 72^{\circ}\)
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