Algebra Know-It-ALL

which simplifies to x2/4. The fraction 2/4 can be reduced to 1/2. That means we actually

have x raised to the 1/2 power, or the square root of x.

10. Imagine that y is a positive number. We take the 6th power of y, and then take the cube
    root or 1/3 power of the result. That gives us

(y^6 )1/3

According to the GMOE rule, that is the same as

y^6 ×(1/3)

which simplifies to y6/3 and then reduces to y^2.

Chapter 9

1. We can suspect that quantity (d), 271/2, is irrational. It is not a natural number. When
   we use a calculator to evaluate it, we get 5.196... followed by an apparently random
   jumble of digits. That suggests its decimal expansion is endless and nonrepeating. The
   other three quantities can be evaluated and found rational:
   (a) 163/4= (161/4)^3 = 23 = 8
   (b) (1/4)1/2= 1/(41/2)= 1/2
   (c) (−27)−1/3= 1/(− 27 1/3)= 1/(−3)=−1/3


2. If we have an irrational number expanded into endless, nonrepeating decimal form,
   we can multiply it by any natural-number power of 10 and always get the same string
   of digits. The only difference will be that the decimal point moves to the right by one
   place for each power of 10. As an example, consider the square root of 7, or 71/2. Using
   a calculator with a large display, we see that this expands to

7 1/2= 2.64575131106459059...

As we multiply by increasing natural-number powers of 10, we get this sequence of

numbers, each one 10 times as large as the one above it:

10 × 7 1/2= 26.4575131106459059...

100 × 7 1/2= 264.575131106459059...

1,000 × 7 1/2= 2,645.75131106459059...

10,000 × 7 1/2= 26,457.5131106459059...

↓

and so on, as long as we want

These are all endless non-repeating decimals, so they’re all irrational numbers. This will

happen for any endless non-repeating string of digits.

Chapter 9 615