Simplify each of the following:
1. 16*(2^x)+2
2. 3[sqrt(x^1/4)*sqrt(x)/5*sqrt(x^2)]
3. ab^(2x+y)/a^x b^x
since 16=2^4, 2^(x+4)+2
sqrt(x^1/4) = x^1/2, so you have 3x/5x = 3/5
assuming the usual carelessness with parentheses, I see
(ab)^(2x+y)/(ab)^x = (ab)^(x+y)
To simplify each of the given expressions, let's go step by step:
1. To simplify 16*(2^x) + 2:
First, we can simplify the exponent by using the property of exponents that says a^(m+n) = a^m * a^n. So we rewrite 16*(2^x) as (2^4)*(2^x), which equals 2^(4+x). Therefore, the expression becomes 2^(4+x) + 2.
2. To simplify 3[sqrt(x^1/4)*sqrt(x)/5*sqrt(x^2)]:
Let's break it down:
- First, simplify the exponent of x within the square root. Notice that x^(1/4) can be rewritten as the fourth root of x, so sqrt(x^(1/4)) becomes the fourth root of x.
- Next, simplify the square root of x^2, which is just x since the square root and square cancel each other out.
- Now, simplify the expression inside the brackets: sqrt(x) / (5 * x).
- Finally, multiply the above expression by 3, resulting in 3 * (sqrt(x) / (5 * x)).
3. To simplify ab^(2x+y)/a^x * b^x:
Let's break it down:
- The division of a^x in the numerator and the exponent b^x in the denominator can be simplified as multiplying a^x by b^(-x). This gives us ab^(2x + y) * a^x * b^(-x).
- Now, we can use the property of exponents that states a^m/a^n = a^(m-n) to simplify the fraction and combine the terms. Therefore, the expression becomes ab^(2x + y - x) = ab^(x + y).