Question

Use the rules for raising a power to a power to explain why raising a number to the power of ½ is the same as finding the square root of the number.

a) Because (a^b)^c = a^(b*c)

b) Because a^(1/b) = √a

c) Because (a^b)^c = a^(b+c)

d) Because a^(1/b) = a^(b-1)

Answer 1

Raising a number to the ½ power is equivalent to taking its square root because of the exponent rule (a^b)^c = a^(b*c), where multiplying the base number by itself with power ½ twice results in the base number to the power of 1, essentially a square root.

Step-by-step explanation:

The rules for raising a power to a power explain why raising a number to the power of ½ is the same as finding the square root of the number. Looking at option (a) Because (a^b)^c = a^(b*c), we can apply this rule to fractional exponents. For instance, when we raise a number to the ½ power, we are essentially multiplying that number by itself to the power of ½ twice, which is equivalent to taking the square root. Using 5 as an example, 5½ . 5½ leads to (5. 5)½ which is 5², and this simplifies to √5. Another way to think about it is that to 'undo' a square, we need to raise it to the power that makes it 1 when multiplied by 2. The power that does this is ½ because 2*½ = 1.