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Zorluk: ZorClassification of Numbers

Consider the following 88 mathematical expressions:

I. π−227\pi - \frac{22}{7}
II. 273\frac{\sqrt{27}}{\sqrt{3}}
III. (3−2)2(\sqrt{3} - \sqrt{2})^2
IV. The infinite decimal 0.101101110…0.101101110\dots (where the number of consecutive 11 s increases by one each time)
V. 1.4141.414
VI. 10×2.5\sqrt{10} \times \sqrt{2.5}
VII. e0e^0
VIII. 2+8\sqrt{2} + \sqrt{8}

How many of the above expressions evaluate to a rational number?

Cevap: 4

Cevap

Exactly 4 of the expressions evaluate to a rational number.
Out of the 8 expressions provided, exactly 4 result in rational numbers: Expression II (simplifies to 3), Expression V (terminating decimal 1414/1000), Expression VI (simplifies to sqrt(25) = 5), and Expression VII (simplifies to 1). The remaining 4 expressions yield irrational results.

Adım Adım Çözüm

1
Evaluate expression I (pi minus 22/7)
The expression evaluates to an irrational number.
The number pi is irrational, while 22/7 is a rational approximation. The difference between an irrational number and a rational number is always irrational.
2
Evaluate expression II (sqrt(27) / sqrt(3))
The expression simplifies to 3, which is a rational number.
Using the properties of radicals, sqrt(27) / sqrt(3) = sqrt(27/3) = sqrt(9) = 3.
3
Evaluate expression III (square of (sqrt(3) - sqrt(2)))
The expression expands to 5 - 2*sqrt(6), which is an irrational number.
Applying the binomial square formula (a-b)^2 = a^2 - 2ab + b^2 yields 3 - 2*sqrt(6) + 2. Since sqrt(6) is irrational, the entire expression is irrational.
4
Evaluate expression IV (the pattern decimal 0.101101110...)
The expression is an irrational number.
The decimal is non-terminating and non-periodic (the pattern changes constantly by adding an extra 1), which is the definition of an irrational decimal.
5
Evaluate expression V (1.414)
The expression is a rational number.
1.414 is a terminating decimal. Any terminating decimal can be written as a fraction of integers (1414/1000), making it rational.
6
Evaluate expression VI (sqrt(10) * sqrt(2.5))
The expression simplifies to 5, which is a rational number.
Multiplying the terms inside the radicals gives sqrt(10 * 2.5) = sqrt(25) = 5.
7
Evaluate expression VII (e^0)
The expression simplifies to 1, which is a rational number.
Any non-zero real number raised to the power of 0 equals 1.
8
Evaluate expression VIII (sqrt(2) + sqrt(8))
The expression simplifies to 3*sqrt(2), which is an irrational number.
sqrt(8) can be simplified to 2*sqrt(2). Adding sqrt(2) gives 3*sqrt(2), which remains an irrational product.
9
Count the total number of rational expressions
Expressions II, V, VI, and VII are rational. Total count is 4.
Identifying the rational outcomes from the previous steps.

Anahtar Kavram

Properties and Definitions of Rational and Irrational Numbers
Tahmini Süre:2m 30s
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Konular

Classification of Numbers

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