Radicals - Rational and irrational numbers: Level 2 The definition of the square root radical
Example 4. Solve for
In other words, if we call In this equation --
Problem 8. Solve for To see the answer, pass your mouse over the colored area.
The definition of the square root radical The square root of A square root radical multiplied by itself That formal rule is the implicit definition of the symbol .
Problem 9. Evaluate the following.
Example 6. Multiply out ( + ). That is, distribute .
Problem 10. 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.
The denominator is now rational.
For we can write any fraction as the numerator sime the reciprocal of the denominator. Finally, rationalizing the denominator simplifies the task of evaluating the fraction. Since we know that , for example, is approximately 1
Problem 12. Show each of the following by transforming the left-hand side.
Real numbers A real number is what we call any rational or irrational number. It is a number we expect to find on the number line. The real numbers are the subject of calculus and of scientific measurement. A real variable is a variable that takes on real values. The term real number was coined by René Descartes in 1637. It was to distinguish it from an imaginary or complex number
Problem 13. Let For example, 3.1, 3.14, , , . Problem 14. 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 Therefore, what values are permitted to the real variable a)
b)
1 + c)
1 − d)
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