1/16(x)^3a + 1/2(y)^(6a)(z)^(9b)
Factor.
= (1/16)(x^(3a) + 8y^(6a)z^(9b) )
= (1/16) [ (x^a)^3 + (2(y^(2a)^3z^(3b) )^3 )
recall A^3 + b^3 = (A+B)(A^2 + AB + B^2)
using A = x^a , B = 2y^(2a)z^(3b)
factor the last part as a sum of cubes
The expression you provided is 1/16(x)^3a + 1/2(y)^(6a)(z)^(9b). To simplify this expression, let's break it down step by step.
Step 1: Simplify the first term
The first term is 1/16(x)^3a. To simplify it further, we can use the properties of exponents. Remember that when you raise a power to another power, you need to multiply the exponents.
So, (x)^3a can be rewritten as (x^3)^a. Multiplying the exponents, we get x^(3a).
Now, the first term becomes 1/16(x^(3a)).
Step 2: Simplify the second term
The second term is 1/2(y)^(6a)(z)^(9b). Let's focus on the exponents first. When you have a product with an exponent, you can distribute that exponent to each factor within the parentheses.
So, (y)^(6a)(z)^(9b) becomes (y^6a)(z^9b).
Now, the second term becomes 1/2(y^6a)(z^9b).
Step 3: Combine the terms
Now that we have simplified each term, we can add them together. Remember that when adding fractions, you need to have a common denominator.
In this case, the common denominator is 16. Multiply the first term by (2/2) to give it a denominator of 16.
The expression now becomes (2/32)(x^(3a)) + 1/2(y^6a)(z^9b).
Step 4: Further simplification
You can simplify the coefficients by dividing both the numerator and denominator by their greatest common divisor. In this case, the GCD of 2 and 32 is 2.
The expression becomes (1/16)(x^(3a)) + 1/2(y^6a)(z^9b).
And that's the simplified form of the expression 1/16(x)^3a + 1/2(y)^(6a)(z)^(9b).