Radicals - Rational and irrational numbers: Level 2
Example 4. Solve for x:
In other words, if we call z the argument of the equation z² = a, then in the solution, the argument is on the left. The argument is whatever was squared.
In this equation --
Problem 7. Solve for x.
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The definition of the square root radical
Here is the formal rule that implicitly defines the symbol :
A square root radical multiplied by itself
Problem 8. Evaluate the following.
Example 6. Multiply out ( + ). That is, distribute .
Problem 9. Following the previous Example, multiply out
( + ).
Rationalizing a denominator
Rationalizing a denominator is a simple technique for changing an irrational denominator into a rational one. We simply multiply the radical by itself. But then we must multiply the numerator by the same number.
Solution. Multiply both the numerator and denominator by :
The denominator is now rational.
reciprocal of the denominator.
Finally, rationalizing the denominator simplifies the task of evaluating the fraction. Since we know that , for example, is approximately 1.414, then we can easily know that
Problem 11. Show each of the following by transforming the left-hand side.
A real number is distinguished from an imaginary or complex number. It is what we call any rational or irrational number. It is a number we expect to find on the number line. It is a number we need for measuring.
The real numbers are the subject of calculus and of scientific measurement.
A real variable is a variable that takes on real values.
Problem 12. Let x be a real variable, and let 3 < x < 4. Name five values that x might have.
For example, 3.1, 3.14, , , .
Problem 13. If the square root is to be a real number, then the radicand may not be negative. (There is no such real number, for example, as .)
If is to be real, then we must have x ≥ 0.
(If you are not viewing this page with Internet Explorer 6 or Firefox 3, then your browser may not be able to display the symbol ≥, "is greater than or equal to"; or ≤, "is less than or equal to.")
Therefore, what values are permitted to the real variable x ?
a) x − 3 ≥ 0; that is, x ≥ 3.
b) 1 + x ≥ 0; x ≥ −1.
c) 1 − x ≥ 0; −x ≥ −1, which implies x ≤ 1.
d) x² ≥ 0. In this case, x may be any real number.
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