### Video Transcript

By considering square numbers,
order the square root of 19, the square root of 24, the square root of 28, four, the
square root of 17, five, and 4.5 from least to greatest.

In this question, we have a mixture
of rational and irrational numbers that we need to order from least to greatest. This is known as ascending
order. We begin by ordering the three
rational numbers. Four is less than 4.5, which is
less than five. Since 17, 19, 24, and 28 are not
square numbers, the square roots of these numbers would give us irrational
numbers.

Next, we recall that if we have two
positive numbers such that 𝑎 is less than 𝑏, then the square root of 𝑎 is less
than the square root of 𝑏. As such, we can order the four
irrational numbers. The square root of 17 is less than
the square root of 19, which is less than the square root of 24, which is less than
the square root of 28.

We now need to compare the
irrational numbers or radicals to the three rational numbers. One way to do this is to rewrite
each rational number as a radical. Since four squared is equal to 16,
we know that four is equal to the square root of 16. Likewise, since five squared is 25,
the square root of 25 is equal to five. We are now left with 4.5, which is
equal to the square root of 4.5 squared. Noting that 4.5 is equal to the
improper or top-heavy fraction nine over two, we can calculate 4.5 squared by
squaring nine over two. We square the numerator and
denominator separately, giving us 81 over four, which is equal to the decimal
20.25. 4.5 is therefore equal to the
square root of 20.25.

We can now order the seven radical
expressions. The smallest is the square root of
16. This is followed by the square root
of 17, the square root of 19, the square root of 20.25, the square root of 24, the
square root of 25, and finally the square root of 28. Replacing the three radical
expressions with their original rational equivalents gives us the set of seven
numbers from least to greatest. In ascending order, we have four,
the square root of 17, the square root of 19, 4.5, the square root of 24, five, and
the square root of 38.