McGraw Hill Glencoe Algebra 2, 2012
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McGraw Hill Glencoe Algebra 2, 2012 View details
5. Operations with Radical Expressions
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Exercise 64 Page 420

You need to find three positive whole numbers and such that

Example Solution:

Practice makes perfect
We need to find a number such that its square root, cube root, and fourth root are positive whole numbers.
In the expressions above, and are positive whole numbers. Using the definition of the root of a number, we obtain the following relations.
Notice that is a number that can be written as three different powers.
Additionally, must be multiplied by itself times to get the value of while and require fewer multiplications by themselves. This leads us to the following relation.
We can also set the following identities.
Since is the smallest value, we will substitute some values for and calculate After that, our mission will be to find the numbers and
Next, we will try to write as a whole number cubed and as another whole number squared.

Since cannot be written as a whole number cube, cannot be Let's make a table where we try some more numbers.

No way
No way
No way
No way
No way

From the table above, one number that satisfies the required condition is

Alternative Solution

Using Rational Exponents
We need to find a number such that its square, cube, and fourth roots are positive whole numbers.
Let's write the number we are looking for as where and are positive whole numbers. Next, we will rewrite the roots as rational exponents.
Here the key is the indexes of the roots, or equivalently, the denominators of the exponents: and Since is a whole number, we need each exponent to be a whole number too.
From the above, we need to be divisible by and One number that satisfies this condition is the least common multiple (LCM) of these numbers.
Consequently, any whole number raised to will have a positive whole number for a square, cube, and fourth root. We can check it by trying some values.

Therefore, any number raised to the power will satisfy the given condition. Particularly, we can pick