Question

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

Answer 1

Answer:

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

Step-by-step 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:

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