ADVANCED EXERCISES - Basic Rules of Algebra

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Prove that each of the following statements holds:

(1) For two real numbers x and y, x² = y² implies x = y or x = -y.

( Click On HINT To Get Some Help , Click On SOLUTION To See The Answer )

(2) For real numbers a, b, c, d with b, d 0,

( ^a/_b ) + ( ^c/_d ) = ^{( ad + bc )}/_bd .

Note: ^x/_y denotes x( ¹/_y ).

( HINT , SOLUTION )

(3) If 0 a < b, then a² < b². ( HINT , SOLUTION )

DEFINITION: If a > 0, then there is a unique b > 0 such that bb = a. We call b "the square root of a" and denote it by .

(4) If x > 0 and y > 0, then

^{( x + y )}/₂ .

( HINT , SOLUTION )

NOTE: is called the geometric mean of x and y, while ^{( x + y )}/₂ is called the arithmetic mean. The inequality in (4) is a famous relation between these two means. One way of thinking about these means is to consider a rectangle R with sides of length x and y. If you create a square with the same AREA as R, then it will have sides of length SQRT(xy), the geometric mean of x and y. If you create a square with same perimeter as R, then it will have sides of length ^{( x + y )}/₂, the arithmetic mean.

(5) Expand: (3x + 4)(x² - 4x + 9)

(SOLUTION )

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