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Chapter 1: Problem 23

Convert to expressions with rational exponents. \(\frac{1}{\sqrt[3]{t^{4}}}\)

Short Answer

Expert verified
The expression with rational exponents is \(t^{-4/3}\).

Step by step solution

01

Identify the Expression

The given expression is \(\frac{1}{\root{3}{t^{4}}}\).
02

Understand the Cube Root in the Denominator

Recognize that \(\root{3}{t^{4}} \) is a cubic root. A cube root can be expressed with a rational exponent as \((t^{4})^{1/3}\).
03

Apply Properties of Exponents

Use the properties of exponents to combine them: \((t^{4})^{1/3} = t^{4 \times 1/3} = t^{4/3}\).
04

Apply the Negative Exponent

Since the term is in the denominator, it needs to be moved to the numerator by changing the exponent to negative: \(\frac{1}{t^{4/3}} = t^{-4/3}\).

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Key Concepts

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

Exponent Rules
Exponent rules are tools that help simplify expressions involving powers. An exponent shows how often a number, termed the base, is multiplied by itself.

Let's break down some key exponent rules:
  • Product of Powers Rule: When multiplying two powers with the same base, add their exponents: \(a^m \times a^n = a^{m+n}\)
  • Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents: \( \frac {a^m} {a^n} = a^{m-n} \)
  • Power of a Power Rule: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{mn}\)
  • Negative Exponent Rule: A negative exponent means you take the reciprocal of the base and change the exponent to positive: \(a^{-n} = \frac {1} {a^n}\)
  • Zero Exponent Rule: Any non-zero base raised to the power of zero is 1: \(a^0 = 1\)
These rules are essential when working with algebraic expressions.
Cube Roots
Cube roots are used to find a number that, when multiplied by itself three times, gives the original number. The cube root of a number \(x\) is written as \(\sqrt[3]{x}\).

In the context of exponents:
  • Rational Exponents: A cube root can be expressed as a rational exponent. For example, \(\sqrt[3]{x} = x^{1/3} \).
  • Property: If there is a power inside the cube root, you multiply the exponents when converting. For instance, \((t^4)^{1/3} = t^{4 \times 1/3} = t^{4/3}\).
Understanding cube roots and their conversion to rational exponents is critical in simplifying expressions.
Negative Exponents
Negative exponents indicate that the base should be moved to the other side of the fraction line. More specifically:

  • A negative exponent in the numerator moves the base to the denominator: \a^{-n} = \frac {1}{a^n}\
  • A negative exponent in the denominator moves the base to the numerator: \ \frac{1}{a^{-n}} = a^n \
In our exercise:
  • The term \(t^{4/3}\) was initially in the denominator as \ \frac{1}{t^{4/3}}\.
  • Converting the negative exponent: Using the rule, we shift the term to the numerator: \ t^{-4/3}\.
Using negative exponents helps simplify expressions and make them easier to work with.

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Most popular questions from this chapter

a) Show that \(\cos 2 t=\cos ^{2} t-\sin ^{2} t\) b) Show that \(\cos 2 t=2 \cos ^{2} t-1\). (Hint: Use part (a) and a Pythagorean Identity.)

Use a calculator to find the values of the following trigonometric functions. \(\sin (2.3)\)

Solar Radiation. The annual radiation (in megajoules per square centimeter) for certain land areas of the northern hemisphere may be modeled with the equation \({ }^{19}\) \(R=0.339+0.808 \cos l \cos s-0.196 \sin l \sin s\) \(-0.482 \cos a \sin s\) In this equation, \(l\) is the latitude (between \(30^{\circ}\) and \(60^{\circ}\) ) and \(s\) is the slope of the ground (between \(0^{\circ}\) and \(60^{\circ}\) ). Also, \(a\) is the aspect, or the direction that the slope faces. For a slope facing due north, \(a=0^{\circ}\), and for a slope facing south, \(a=180^{\circ} .\) For a slope facing either east or west, \(a=90^{\circ}\). Find the annual radiation of south-facing land at \(30^{\circ}\) north latitude with a \(20^{\circ}\) slope.

Use a calculator to find the values of the following trigonometric functions. \(\cot (3 \pi / 11)\)

Compute \(\sin 105^{\circ} .\) (Hint: Use a Sum Identity and the [act that \(105^{\circ}=45^{\circ}+60^{\circ}\).)
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