logarithm problem?

Question

simplify and leave in radical form. cube root of 5^2.7 divided by the cube root of 5^4.5

the answer is root (index 5) of 5^2 divided by five. any idea how they got there??

Answer 1

(5^2.7)^(1/3) / (5^4.5)^(1/3)

Rule #1:
(a^b)^c = a^bc

(5^0.9) / (5^1.5)

Rule #2
a^b / a^c = a^(b - c)

5^(-0.6)
5^(-6/10)
5^(-3/5)

(5^-3)^1/5

Rule #3:
a^-b = 1 / a^b

(1 / 5^3)^(1/5)

Rule #4:
(a/b)^c = a^c / b^c

1^(1/5)
---------
5^3^(1/5)

1 to any power is 1, so the numerator is 1:
.... 1
-----------
5^3^(1/5)

Now multiply top and bottom by (25^(1/5)). This will give you an even power of 5. (25 x 125 = 5^5):
25^(1/5)
---------------
(5^5)^(1/5)

The bottom becomes 5^(5*1/5) = 5^1 = 5
25^(1/5)
----------
.... 5

That's the simplest form with no radicals in the denominator:
fifth root(25) / 5

Answer 2

Taking the cube root means multiplying the exponent by 1/3 (or dividing by 3).

5^0.9/5^1.5=5^(-0.6)=0.38073~

-0.6 power is the same as -3/5 power. That's the fifth root of the cube of the reciprocal of the number (working from right to left of the fraction).

Answer 3

Here is how:

dividing two cube roots can be done by dividing the numbers under the cube root first.

5^2.7 / 5^4.5 = 5^-1.8 (subtract exponents when dividing)

cube root of 5^-1,8 = (5^-1.8)^1/3

= 5^ (-1.8/3)

= 5^ (-3/5)

= (fifth root of 5) ^ -3

= (fifth root of 5)^(2-5)

= fifth root of 5^2 over fifth root of 5^5 which is 5

Answer 4

We have ((5^2.7)/(5^4.5))^(1/3). We'll say it equals 'A'. if we take a log on base 5 of both sides:
log(((5^2.7)/(5^4.5))^(1/3))=logA where all the logs are base 5. Log to a power equals the power times the log:
=(1/3)*log((5^2.7)/(5^4.5))=(1/3)*log(5^-1.8)=(1/3)*(-1.8)*log5.
since log 5 when the base is 5 equals 1:
=(1/3)*(-1.8)=-0.6
That equals our log of A. logA=-0.6, so according to the definition of a log, the base (5) times -0.6 will equal A, so the answer:
A=5^(-0.6)=1/5^0.6

Logarithms aren't needed though, because the cube root of A dvided by the cube root of B equals the cube root of A/B. so:
A=((5^2.7)/(5^4.5))^(1/3)=(5^-1.8)^(1/3)=5^(-1.8/3)=5^(-0.6)

Answer 5

cbrt(5^2.7)/cbrt(5^4.5)
Cube it.
(------)^3
=5^2.7/5^4.5
The rule for dividing powers says a^m/a^n=a^(m/n). In this case, a=5, m=2.7, and n=4.5. Solving, you get
5^0.6
Since you cubed it, put a cube root over it.

The answer is the cube root of 5^0.6

Answer 6

cube root of 5^2.7=5^0.9
cube root of 5^4.5=5^1.5
5^0.9/5^1.5
=5^(-0.6) or 1/5^(0.6)
or simplycuberoot of 5^(2.7-4.5)
=>cube root of 5^(-1.8)

Answer 7

(5^2.7)^(1/3) = 5^(2.7/3) 5^0.9
(5^4.5)^(1/3) = 5^(4.5/3) = 5^1.5

so it is
5^0.9/5^1.5 = 5^(0.9-1.5) = 5^(-0.6)
= 1/5^(6/10)
= 1/5^(3/5)
= fifth root of (1/5^3)
= fifth root of (1/125)
= fifth root of (0.008)