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Chapter 1: Problem 2

Express the number in the form a/b, where \(a\) and \(b\) are integers. $$(-3)^{3}$$

Short Answer

Expert verified
The number is expressed as \(-\frac{27}{1}.\)

Step by step solution

01

Understand the Problem

The task is to express the number obtained from evaluating (-3)^{3}as a fraction of the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers.
02

Evaluate the Expression

To evaluate \((-3)^{3}\), we need to multiply \(-3\) three times: (-3) \times (-3) \times (-3).Starting with the first two factors, -3 \times -3 = 9.Next, multiply the result by the third factor: 9 \times -3 = -27.
03

Express as Fraction

The result of \((-3)^{3}\) is an integer \(-27\). To express this as a fraction, we can write it with a denominator of 1: \(-27 = \frac{-27}{1}.\)

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Key Concepts

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

Exponents
Exponents are a shorthand used in mathematics to denote repeated multiplication of a base number. In the exercise,
we have the expression \((-3)^3\). Here, \(-3\) is the base and \(3\) is the exponent. This expression means that we multiply \(-3\) by itself three times. Explaining further, think of the exponent as instructions for how many times to use the base as a factor. This is particularly useful when dealing with large numbers or calculations that need repeated multiplication.
  • Base: The number that is multiplied by itself.
  • Exponent: Indicates how many times the base is used as a factor.
  • Example: \((-3)^3 = -3 imes -3 imes -3\).
Exponents simplify mathematical expressions and are foundational in algebra, enabling more efficient calculations.
Integers
Integers are the set of whole numbers and their negatives, including zero. In the given exercise,
we work with the integer \(-3\) raised to the power of 3, resulting in \(-27\), which is also an integer. Integers are important in mathematics because they include both positive and negative numbers, allowing us to express a range of values, including zero.
  • Positive Integers: Numbers greater than zero \((1, 2, 3, ext{etc.})\).
  • Negative Integers: Numbers less than zero \((-1, -2, -3, ext{etc.})\).
  • Zero: Considered an integer as it is neither positive nor negative.
Using integers in calculations provides flexibility and accuracy, especially when dealing with real-world problems like finance or temperature changes.
Fractions
Fractions represent parts of a whole and are composed of two integers: a numerator and a denominator. In this exercise, the result of calculating \((-3)^3\) is the integer \(-27\), which we express as a fraction \(-\frac{27}{1}\). This representation illustrates that any integer can be seen as a fraction by using 1 as the denominator.
  • Numerator: The top part of a fraction representing how many parts we have.
  • Denominator: The bottom part which represents the total number of equal parts.
  • Simplification: Fractions can be simplified by dividing the numerator and the denominator by their greatest common divisor.
Understanding fractions enables precise calculations in various fields and is essential for expressing ratios, proportions, and probabilities.

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Most popular questions from this chapter

Simplify the expression. $$\frac{4 x}{3 x-4}+\frac{8}{3 x^{2}-4 x}+\frac{2}{x}$$

Simplify the expression. $$\left(2 x^{2}-3 x+1\right)(4)(3 x+2)^{3}(3)+(3 x+2)^{4}(4 x-3)$$

In evaluating negative numbers raised to fractional powers, it may be necessary to evaluate the root and integer power separately. For example, \((-3)^{2 / 5}\) can be evaluated successfully as \(\left[(-3)^{1 / 5}\right]^{2}\) or \(\left[(-3)^{2}\right]^{1 / 5}\), whereas an error message might otherwise appear. Approximate the realnumber expression to four decimal places. (a) \((-3)^{2 / 5}\) (b) \((-7)^{4 / 3}\)

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt{\frac{1}{3 x^{3} y}}$$

Choose the equation that best describes the table of data. $$\begin{array}{|c|c|}\hline \boldsymbol{x} & \boldsymbol{y} \\\\\hline 1 & -9 \\\2 & -4 \\\3 & 11 \\\4 & 42 \\\5 & 95 \\\\\hline\end{array}$$ (1) \(y=13 x-22\) (2) \(y=x^{2}-2 x-8\) (3) \(y=4 \sqrt{x}-13\) (4) \(y=x^{3}-x^{2}+x-10\)
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