Question

Find the value of a given ddle" class="latex-formula">=³√5²

Answer 1

Answer:

  a = 2/3

Step-by-step explanation:

A cube root is the same as a 1/3 power. The square of that gives a 2/3 power.

  a = 2/3

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Additional comment

You can see the relationship between a root index and an exponent if you consider what the root means.

Consider a cube root, for example. When you cube the root, you get the original number:

  (∛x)·(∛x)·(∛x) = x

Now, let's write the root as a power of x: x^a.

  (x^a)·(x^a)·(x^a) = x . . . . . where x^a = ∛x

We know this product is ...

  x^(a+a+a) = x^(3a) = x^1

This tells us that ...

  3a = 1   ⇒   a = 1/3

That is, ∛x = x^(1/3).

Of course an n-th root is multiplied by itself n times to get the original number, so the corresponding exponent is x^(1/n).