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Chapter 1: Problem 5

In exercises \(5-7,\) write the given term without using exponents. $$ \left(8 x_{1}-5 x_{2}+11\right)^{-1 / 3} $$

Short Answer

Expert verified
\(\frac{1}{\sqrt[3]{8x_1 - 5x_2 + 11}}\)

Step by step solution

01

Understand the Expression

The expression given is \((8x_1 - 5x_2 + 11)^{-1/3}\). This is an exponent expression, specifically a fractional exponent that indicates a root.
02

Identify the Meaning of the Exponent

The exponent \(-1/3\) means two things: the negative sign indicates taking the reciprocal of the expression, and the \(1/3\) indicates taking the cube root of the expression.
03

Write the Reciprocal

The negative exponent means we take the reciprocal of the expression \((8x_1 - 5x_2 + 11)\). This gives us \(\frac{1}{(8x_1 - 5x_2 + 11)^{1/3}}\).
04

Write as a Root

The exponent \(1/3\) after taking the reciprocal indicates a cube root. Therefore, \((8x_1 - 5x_2 + 11)^{1/3}\) is equivalent to \(\sqrt[3]{8x_1 - 5x_2 + 11}\).
05

Combine the Steps

Substitute the cube root back into the reciprocal expression. The final expression without exponents is \(\frac{1}{\sqrt[3]{8x_1 - 5x_2 + 11}}\).

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Key Concepts

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

Understanding Exponents
Exponents are a shorthand way to express repeated multiplication of the same number. For example, the expression \( a^n \) represents multiplying \( a \) by itself \( n \) times. Exponents make it easier to work with very large or very small numbers, and they help simplify algebraic expressions.
There are different kinds of exponents:
  • Positive whole number exponents indicate multiplying the base by itself the given number of times. For example, \( 2^3 = 2 \times 2 \times 2 = 8 \).
  • Fractional exponents, like \( x^{1/2} \), are a way to indicate roots. For example, \( x^{1/2} \) is the same as \( \sqrt{x} \), which is the square root of \( x \).
  • Negative exponents represent reciprocal values, such as \( x^{-1} = \frac{1}{x} \).
In the exercise provided, we dealt with a fractional exponent \(-1/3\), which involves both a negative and fractional component. This means we first take the reciprocal (due to the negative sign), and then determine the cube root (because of the \( 1/3 \)).
Exploring Cube Roots
Cube roots are special kinds of roots that mean finding a number which, when multiplied by itself three times, gives the original number.
The cube root of a number \( x \) is written as \( \sqrt[3]{x} \) or equivalently as \( x^{1/3} \).
  • For instance, \( \sqrt[3]{8} = 2 \) because \( 2 \times 2 \times 2 = 8 \).
  • Cube roots can be applied not only to numbers but also to algebraic expressions, similar to what we saw in the exercise, where \( (8x_1 - 5x_2 + 11)^{1/3} \) became \( \sqrt[3]{8x_1 - 5x_2 + 11} \).
Cube roots play an essential role in simplifying expressions with fractional exponents, especially when coupled with reciprocals in expressions with negative fractional exponents.
The Role of Reciprocals
Reciprocals are mathematical expressions where the original number is flipped over one.
If you have a number \( a \), its reciprocal is \( \frac{1}{a} \). When applied to fractions, you simply invert the fraction; for example, the reciprocal of \( \frac{b}{c} \) is \( \frac{c}{b} \).
  • Reciprocals are useful when dealing with negative exponents because they translate the problem into a form that's easier to work with. For example, \( a^{-1} \) becomes \( \frac{1}{a} \).
  • In more complex expressions like the one in our exercise, the reciprocal helps to eliminate the negative exponent by placing the expression in the denominator. This results in a much more intuitive form, \( \frac{1}{\sqrt[3]{8x_1 - 5x_2 + 11}} \).
Understanding reciprocals is important, especially when dealing with expressions that involve both negative and fractional exponents.

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Most popular questions from this chapter

In exercises \(18-20,\) simplify and write the given term in exponential form. $$ \sqrt[3]{\left(\frac{e^{4 \theta-6} y^{2}}{e^{\theta} y^{-4}}\right)} $$

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Let \(f(x)=x^{3}, g(x)=x+4,\) and \(h(x)=\sin (x) .\) Each of exercise \(5-8\) is some combination of \(f(x), g(x),\) and \(h(x) .\) Determine the type of combination and write it using function notation. For example, \(x^{3}+x+4\) is the addition of \(f(x)\) and \(g(x)\) and can be written as \(f(x)+g(x)\). $$ \sin (x)+4 $$

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