Simplifying exponents 4(5^(2n+1))-10(5^(2n-1))/ ( 2(5^n))
When factoring always take out the power with the smaller of the exponents
4(5^(2n+1))-10(5^(2n-1))/ ( 2(5^n))
= 2(5^(2n-1) ) ( 2(5^2 - 5(1) )/2(5^n))
= 5^(n-1) (50 - 5)
= 225 (5^(n-1) )
PLease explain furter a little the steps of this...
4(5^(2n+1))-10(5^(2n-1))/ ( 2(5^n))
= 2(5^(2n-1) ) ( 2(5^2 - 5(1) )/2(5^n))
= 5^(n-1) (50 - 5)
= 225 (5^(n-1) )
To simplify the expression 4(5^(2n+1))-10(5^(2n-1))/ ( 2(5^n)), we can use the properties of exponents and simplify each term separately, and then combine like terms if possible.
Let's start with the numerator:
4(5^(2n+1))-10(5^(2n-1))
In the first term, 4 is multiplying 5 raised to the power of (2n+1). Using the property of exponents, raising a power to another power, we can multiply the exponents:
4(5^(2n+1)) = 4(5^2n * 5^1) = 4(25^n * 5) = 4(125^n)
In the second term, 10 is multiplying 5 raised to the power of (2n-1). Using the same property of exponents:
10(5^(2n-1)) = 10(5^2n * 5^(-1)) = 10(25^n * 1/5) = 10(5 * 25^n)/5 = 10(5 * 25^n)/5
Now let's simplify the denominator: 2(5^n)
There is no need to further simplify this expression.
Now, let's put the numerator and denominator together:
(4(125^n) - 10(5 * 25^n)/5) / (2(5^n))
Next, we can simplify the numerator by combining like terms:
(4(125^n) - (50 * 25^n))/5
Finally, we divide the numerator by the denominator:
(4(125^n) - (50 * 25^n))/5 / (2(5^n))
Now the expression is simplified.