Question

Without using calculator, determine between which two integers the following surds lie (d)⁵√30 (e)³√45​

Answer 1

• 30^(1/5) is in (1, 2)
• 45^(1/3) is in (3, 4)

Explanation:

You want to know the consecutive integers that bound the values of 30^(1/5) and 45^(1/3).

Powers and roots

The fifth root of 30 will lie between the two integers whose 5th powers lie on either side of 30:

1^5 = 1 < 30

2^5 = 32 > 30

So, the fifth root of 30 lies between 1 and 2:

`\boxed{1 < \sqrt[5]{30} < 2}`

The cube root of 45 will lie between the two integers whose cubes lie on either side of 45:

3^3 = 27 < 45

4^3 = 64 > 45

So, the cube root of 45 lies between 3 and 4:

`\boxed{3 < \sqrt[3]{45} < 4}`

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Additional comment

A calculator confirms this:

`\sqrt[5]{30}\approx 1.974\\\sqrt[3]{45}\approx3.557`