Question

Have you ever seen rational exponents?

Rational exponents are designed to fit within
the properties of exponents. That is, (31/2)2 is
equal to 31, or 3. What operation does the
power 1/2 represent? Can you think of
operations that other fractional powers
represent, such as 1/3, 1/4 and 2/3?

Answer 1

• 1/2 power = square root
• 1/3 power = cube root
• 1/4 power = fourth root
• 2/3 power is the square of the cube root, or the cube root of the square

Explanation:

If you're going to write exponents in plain text, you need to use the exponentiation symbol, a caret (^). And you need to include parentheses around any exponent that is not a single letter or number. For example, the order of operations will tell you that 3^1/2 is evaluated as (3^1)/2 = 1.5, not the square root of 3.

Your first expression is apparently supposed to be ...

(3^(1/2))^2

The Order of Operations comes into play other ways, too. It tells you the meaning of a^b^c is a^(b^c). That is why the extra parentheses are needed around (3^(1/2)) in the above expression.

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We know that if "a" is the square root of x, then it satisfies ...

(√x)(√x) = a·a = x

or

a^2 = x

If we raise both sides of this equation to the 1/2 power, we see that ...

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

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The rules of exponents tell us that ...

(a^b)^c = a^(bc)

so ...

(a^2)^(1/2) = a^(2/2) = a^1 = a

but we already said

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

so that means ...

a = x^(1/2) = √x

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We can extend this idea to a root of any degree (index). The 5th root is ...

a·a·a·a·a = x

a^5 = x

a = x^(1/5) =
`\sqrt[5]{x}`

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The above rule of exponents tells us that if the rational power has a numerator greater than 1, it represents a power of the root (or a root of the power).

x^(2/3) = (x^2)^(1/3) = (x^(1/3))^2