Problem 1
Answers:
Real = sqrt(49), 5/6, 12.5, 8/12, 225, sqrt(24)/16, sqrt(32)
Not real = sqrt(-20) and sqrt(-16)/9
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Explanation:
Anything involving a square root of a negative number will result in a non-real answer. Anything else is real.
Traditionally y = sqrt(x) has a domain of nonnegative numbers. If x >= 0, then y is some real number. If x < 0, then we'll need to use the imaginary number i = sqrt(-1).
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Problem 2
Answers:
Rational = sqrt(49), 5/8, 12.5, 8/12, and 225
Irrational = sqrt(24)/16, sqrt(32)
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Explanation:
Ignore the non-real numbers. Focus solely on the real values.
A rational number is where we can write it as a ratio of two integers. An example is 5/8 since 5 and 8 are integers. The denominator can never be zero.
Decimal values that terminate can be written as a fraction. For instance, 12.5 = 125/10.
Square roots of perfect squares are also rational. For example sqrt(49) = 7 = 7/1
Something like sqrt(32) is not rational because we can't write it in the form p/q with p,q being integers. We say it is irrational.
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Problem 3
Answers:
Perfect Squares = 225
Non-perfect squares = everything else
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Explanation:
A perfect square is of the form x^2, where x is a positive integer.
The only perfect square listed is 225 because 225 = 15^2. Here x = 15.
Note: a perfect square is an integer itself.
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Problem 4
As discussed earlier, rational numbers are fractions of integers. So we have something of the form p/q where p,q are integers and q is nonzero. An example of a rational number is p/q = 7/8 with p = 7 and q = 8.
Irrational numbers are ones that cannot be written as p/q. A famous example of an irrational number is pi = 3.14, but we could do something like sqrt(17) for instance as well. The stuff under the square root must be a non-perfect square.
Perfect squares are integers in the form x^2, with x being an integer also. A visual way to think of perfect squares is to think of a square with side length x, so its area is x^2. If x = 9, then its area is x^2 = 9^2 = 81, so 81 is a perfect square. We only consider whole number side lengths.
