Algebra Know-It-ALL

138 Irrational and Real Numbers

Quotient of quotients

For all real numbers w, x, y, and z where x≠ 0, y≠ 0, and z≠ 0,

(w /x)/(y /z)= (w /x)(z /y)

= (w /y)(z /x)

= (wz)/(xy)

Sum of quotients

For all real numbers w, x, y, andz where x≠ 0 and z≠ 0,

w /x+y /z= (wz+xy)/(xz)

Integer roots

Suppose that x is a positive real number. Also suppose that n is a positive integer. Then the nth

root of x can also be expressed as the 1/n power of x. The second root (or square root) is the

same thing as the 1/2 power, the third root (or cube root) is the same thing as the 1/3 power,

the fourth root is the same thing as the 1/4 power, and so on.

Rational-number powers

Suppose that x is a real number. Also suppose that m and n are integers, and n≠ 0. Then

xm /n= (xm)1/n= (x1/n)m

Negative powers

Letx be a nonzero real number. Let y be any real number. Then

x−y= (1/x)y= 1/(xy)

Sum of powers

For all nonzero real numbers x, y, and z,

x(y+z)=xyxz

Difference of powers

For all nonzero real numbers x, y, andz,

x(y−z)=xy/xz

Product of powers

For all nonzero real numbers x, y, and z,

xyz= (xy)z= (xz)y