Manipulating exponents

[bpdulog]

I've seen a few exponents being manipulated in the following manner, but can't figure out how this is performed. Can anyone break these examples down in more detail?

1. sqrt((5^2)^3)) translates to 5^3
2. 2x^-4 translates to 2/x^4

[vineeshp]

1. sqrt((5^2)^3)) translates to 5^3

Lets see how the exponent itself can be manipulated.

(5^2)^3 = (5^2) * (5^2) * (5^2) (if you don't agree, then stop at this point and spend some more time understanding exponents.

(5^2) * (5^2) * (5^2) = (5*5) * (5*5) * (5*5) = 5 * 5 * 5 * 5 * 5 * 5 = (5)^6
At this point note that (5)^6 can also be written as ((5^3)^2)

sqrt((5)^6) = 5^3
(For eg, sqrt(25) = sqrt(5^2) = 5 i.e. the powers become half. Agree?)

So an easier way of doing this?
Now how is sqrt denoted in powers? 1/2. agree? So
sqrt((5^2)^3)) = sqrt((5^3)^2)) = ((5^3)^2)^1/2)
here 2 and 1/2 cancel each other ( product of 2 * 1/2 is 1)

((5^3)^2)^1/2) = 5^3

[vineeshp]

2. 2x^-4 translates to 2/x^4

Well that is by convention, rules of exponents.

For eg. 5 = 5^1 and 1/5 = 1/(5^1) = 5^(-1)
Similarly

x^-4 = 1/x^4.

There is nothing much I can explain here. Hope you understood.

[neelgandham]

1. sqrt((5^2)^3)) translates to 5^3

Property :for all real numbers a, b, and c,
(a^b)^c = a^(bc)

So, the expression sqrt((5^2)^3)) = ((5^2)^3))^0.5 = 5^(2*3*0.5) = 5^3

2. 2x^-4 translates to 2/x^4

Property : for all real numbers a, n,
(a^-n) = a^n

So, the expression 2*x^-4 = 2*(1/x^4) = 2/(x^4)
FOr all properties of exponents : https://www.algebralab.org/studyaids/stu ... a2_6-1.xml

[bpdulog]

2. 2x^-4 translates to 2/x^4

Well that is by convention, rules of exponents.

For eg. 5 = 5^1 and 1/5 = 1/(5^1) = 5^(-1)
Similarly

x^-4 = 1/x^4.

There is nothing much I can explain here. Hope you understood.

Shouldn't 2x^-4 translate to 1/2x^4? You're just taking the negative reciprocal right?

[neelgandham]

Shouldn't 2x^-4 translate to 1/2x^4? You're just taking the negative reciprocal right?

Hey bpdulog !

You are correct, it is just the way you look

Case 1: 2*(x^-4) = 2*((1/x)^4) = 2/(x^4)
Case 2: (2x)^-4 = 1/((2x)^4).

These are two different expressions !