Question

Order cube root of eighty-eight, twenty-eight ninths, square root of nineteen from greatest to least.

cube root of eighty-eight, twenty-eight ninths, square root of nineteen
twenty-eight ninths, square root of nineteen, cube root of eighty-eight
twenty-eight ninths, cube root of eighty-eight, square root of nineteen
cube root of eighty-eight, square root of nineteen, twenty-eight ninths

Answer 1

the ordering is

28/9 < 19³ < 88² ⇒ 28/9 < √19 < ∛88

Explanation:

Answer 2

(a) twenty-eight ninths, square root of nineteen, cube root of eighty-eight

Explanation:

When ordering a list of numbers by hand, it is convenient to convert them to the same form. Decimal equivalents are easily found using a calculator.

Order

The attachment shows the ordering, least to greatest, to be ...

`(28)/(9).\ √(19),\ \sqrt[3]{88}`

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Additional comment

We know that √19 > √16 = 4, and ∛88 > ∛64 = 4, so the fraction 28/9 will be the smallest. That leaves us to compare √19 and ∛88, both of which are near the same value between 4 and 5.

One way to do the comparison is to convert these to values that need to have the same root:

√19 = 19^(1/2) = 19^(3/6) = sixthroot(19³)

∛88 = 88^(1/3) = 88^(2/6) = sixthroot(88²)

The roots will have the same ordering as 19³ and 88².

Of course, these values can be found easily using a calculator, as can the original roots. By hand, we might compute them as ...

19³ = (20 -1)³ = 20³ -3(20²) +3(20) -1 = 8000 -1200 +60 -1 = 6859

88² = (90 -2)² = 90² -2(2)(90) +2² = 8100 -360 +4 = 7744

Then the ordering is ...

28/9 < 19³ < 88² ⇒ 28/9 < √19 < ∛88