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2 years ago

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Explain why the square root of a number is defined to be equal to that number to the 1/2 power

1 answer:

Snezhnost [94]2 years ago

8 0

9514 1404 393

Answer:

(x^(1/2))(x^(1/2)) = x^(1/2 +1/2) = x^1 = x

Step-by-step explanation:

The rule of exponents is ...

(x^a)(x^b) = x^(a+b)

From which ...

(x^a)(x^a) = x^(a+a) = x^(2a)

So, if we want two identical factors that have a product of x = x^1, then the exponents of those factors will be such that ...

x^(2a) = x^1

2a = 1

a = 1/2

The square root is defined as one of two identical factors that have a product equal to the specified value. That is ...

(√x)(√x) = x

Above, we have shown that ...

(x^(1/2))(x^(1/2)) = x

so, we can conclude ...

√x = x^(1/2)

_____

<em>Additional comment</em>

In like fashion, we can show that the n-th root of a number is the same as that number to the 1/n power. It's really a matter of definition. Since the square of x^(1/2) is x, we call x^(1/2) the square root. It is used commonly enough that it has its own symbol: √x.

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