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Chapter 0: Problem 81

Which of the following are rational and which are irrational? (a) \(-\sqrt{9}\) (b) 0.375 (c) \((3 \sqrt{2})(5 \sqrt{2})\) (d) \((1+\sqrt{3})^{2}\)

Short Answer

Expert verified
(a) Rational, (b) Rational, (c) Rational, (d) Irrational

Step by step solution

01

Evaluate the Square Root

First, we evaluate \(-\sqrt{9}\). The square root of 9 is 3, so \(-\sqrt{9} = -3\). Since \(-3\) is an integer, and integers are rational numbers, \(-\sqrt{9}\) is a rational number.
02

Check Decimal Representation

Next, let's consider 0.375. A decimal number is rational if it can be expressed as a fraction of two integers. For 0.375, it can be written as \(\frac{375}{1000}\). By simplifying, we get \(\frac{3}{8}\), which is a fraction, so 0.375 is a rational number.
03

Simplify the Product of Square Roots

Now, for \((3 \sqrt{2})(5 \sqrt{2})\), we apply the formula for the product of square roots: \((a \sqrt{b})(c \sqrt{b}) = acb\). Therefore, \(3 \cdot 5 \cdot 2 = 30\). Since 30 is an integer and integers are rational, \((3 \sqrt{2})(5 \sqrt{2})\) is rational.
04

Expand the Binomial Expression

For \((1+\sqrt{3})^{2}\), expand using the formula \((a+b)^{2} = a^2 + 2ab + b^2\). Substituting our terms: \(1^2 + 2 \cdot 1 \cdot \sqrt{3} + (\sqrt{3})^2 = 1 + 2\sqrt{3} + 3 = 4 + 2\sqrt{3}\). The term \(2\sqrt{3}\) is irrational, so the entire expression \(4 + 2\sqrt{3}\) is irrational.

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Key Concepts

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

Square Roots
Square roots are special mathematical operations where a number is multiplied by itself to return the original value. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9. We denote this operation as \(\sqrt{9} = 3\).
Square roots can also yield irrational numbers. An irrational number cannot be expressed as a simple fraction;
instead, it has a non-repeating, non-terminating decimal expansion. For instance, \(\sqrt{2}\) is an irrational number.
  • When evaluating an expression like \(-\sqrt{9}\), we perform the square root operation first, which gives us 3, and then apply the negative sign, resulting in -3.
  • This result is rational, since -3 is an integer.
Square roots are pivotal in determining the irrationality and rationality of numbers. Calculating them is often a crucial first step in any problem involving these types of numbers.
Fraction Representation
A number is considered rational if it can be represented as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b\) is not zero.
Take the decimal 0.375, for example. To convert it into a fraction, you would count the number of decimal places and multiply by a corresponding power of 10 to clear the decimal, thus obtaining \(\frac{375}{1000}\).
After simplifying this fraction, we achieve \(\frac{3}{8}\), which evidences its rational nature.
  • To simplify \(\frac{375}{1000}\), look for the greatest common divisor of 375 and 1000, which is 125.
  • Divide both the numerator and denominator by this number to get the simplified fraction \(\frac{3}{8}\).
Such techniques of fraction representation work not only for integers but for any decimal number that can be cleanly converted into a fraction.
Binomial Expansion
Binomial expansion involves expanding expressions that are raised to a power, using the formula \(\left(a + b \right)^{n}\).
This is a fundamental concept in algebra. Let's explore the expression \((1+\sqrt{3})^{2}\).
When we expand \((a+b)^{2}\), we get \(a^2 + 2ab + b^2\), applying it to our terms, it becomes:
  • \(\left(1+\sqrt{3}\right)^{2} = 1^{2} + 2 \cdot 1 \cdot \sqrt{3} + \left(\sqrt{3}\right)^{2} = 1 + 2\sqrt{3} + 3\)
  • The expression simplifies further to \(4 + 2\sqrt{3}\).
The term \(2\sqrt{3}\) is irrational because \(\sqrt{3}\) does not simplify to a rational number.
Thus, the result \(4 + 2\sqrt{3}\) retains its irrational property due to the presence of the irrational component.

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Most popular questions from this chapter

Sketch the graph of \(g(x)=(x+1)^{3}-3\) using translations.

Sketch the graphs of \(f(x)=(x-3) / 2\) and \(g(x)=\sqrt{x}\) using the same coordinate axes. Then sketch \(f+g\) by adding \(y\) -coordinates.

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(x=-4 y^{2}-1\)

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(3 x^{2}+4 y^{2}=12\)

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(7 x^{2}+3 y=0\)
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