Simplify
5^-1/2 * 5x^5/2 (5x)^-3/2
My final answer was x/sqrt5
you are correct.
x^5/2 * x^-3/2 = x^1 = x
5^-1/2 * 5^1 * 5^-3/2 = 5^-1 =1/5
x/5
I disagree, if you interpret the expression as
5^-1/2 * (5x)^5/2 * (5x)^-3/2
as he apparently did. I must say though, in Damon's defense, Stephen's wily (wildly?) and inconsistent use of parentheses left things open to misinterpretation.
Actually, even that is wrong, because you then have
5^-1/2 * 5x = x√5, not x/√5
So, my bad, too!
I'm just trying to write it like it appears in the book.
(5)^-1/2 * (5x)^5/2 (5x)^-3/2
Simplifies to
1/5 * �ã5x^5(1/�ã5x^3)
How do I get to x�ã5?
I'm confused. I've tried many times.
To simplify the expression, we can use the properties of exponents. Let's break down each term separately:
1. Simplifying 5^-1/2:
The negative exponent indicates that we need to find the reciprocal of the base. So, 5^-1/2 can be written as 1/5^1/2 or 1/sqrt(5).
2. Simplifying 5x^5/2:
Here, we use the rule of exponents that states (a^m)^n = a^(m*n). Applying this to (5x)^-3/2, we have (5x)^(5/2 * -3/2). Multiplying the exponents gives us (5x)^(-15/4).
Now, let's combine these simplified terms:
(1/sqrt(5)) * (5x)^(-15/4)
To make the multiplication easier, let's convert (5x)^(-15/4) into a positive exponent:
(1/sqrt(5)) * [(1/(5x)^(15/4))]^-1
Now, we can simplify the expression further by combining the fractions and shifting the negative exponent:
(1 * (5x)^(15/4))/(sqrt(5))
Finally, let's rewrite (5x)^(15/4) using rational exponents:
(1 * (5)^(15/4) * (x)^(15/4))/(sqrt(5))
Simplifying the numerical part, we have:
(5^(15/4) * x^(15/4))/(sqrt(5))
Since we want to write the final answer as x/sqrt(5), we rewrite the expression by separating x and sqrt(5) into different fractions:
(x^(15/4))/(sqrt(5)^(15/4)) * (5^(15/4))/(sqrt(5)^(15/4))
Now, notice that sqrt(5)^(15/4) is the same as (sqrt(5))^((15/4)*(4/4)), which simplifies to (sqrt(5)^15)/(sqrt(5)^4). The square root raised to the 4th power cancels out, leaving just sqrt(5)^15.
This gives us:
(x^(15/4))/(sqrt(5)^15) * (5^(15/4))/1
Now, simplify the expression by canceling out sqrt(5)^15 in both the numerator and the denominator:
(x^(15/4))/1 * (5^(15/4))/1
Multiplying the terms together:
(x^(15/4) * 5^(15/4))/1
Finally, we can combine the powers of x and 5:
(x*5)^(15/4)
And simplifying further:
(5x)^(15/4)
Therefore, your final answer is (5x)^(15/4).