How do I simplify the following problems? Assume that no denominator equals 0.
1) 8y^3+27/2xy-10y+3x-15
2) sqrt16x^2y^4
3) 2+i/1-3i
To simplify the given expressions, we'll go through each problem step by step:
1) 8y^3 + 27 / 2xy - 10y + 3x - 15
This expression consists of multiple terms combined together. To simplify, we need to group like terms and combine them.
First, let's look for terms with common factors. In this case, we have the term "27" and "15" which can both be factored as "3" times "3" and "5," respectively. So, we can rewrite the expression as:
8y^3 + (3)(3) / 2xy - 10y + 3x - (3)(5)
Next, let's rearrange the terms so that they are in a more organized form:
8y^3 - 10y + 3x + (3)(3) / 2xy - (3)(5)
Now, it's important to note that we cannot combine the terms "8y^3" and "10y" since they have different powers of "y." However, we can combine the constants "9" and "15" to get "24"). Additionally, we can rewrite the expression as:
8y^3 - 10y + 3x + 9 / 2xy - 15
So, the simplified expression is:
8y^3 - 10y + 3x + 9 / 2xy - 15
2) sqrt(16x^2y^4)
To simplify the given expression, we can apply the properties of exponents.
The square root of a number "a" raised to the power of "n" can be written as "a^(n/2)."
In this case, we have sqrt(16x^2y^4) which can be rewritten as (16x^2y^4)^(1/2).
To simplify further, we can apply the exponent rule by multiplying the exponents inside the parentheses with the exponents outside the parentheses:
16^(1/2) * (x^2)^(1/2) * (y^4)^(1/2)
Now, let's simplify each term within the parentheses:
16^(1/2) = sqrt(16) = 4
(x^2)^(1/2) = sqrt(x^2) = x
(y^4)^(1/2) = sqrt(y^4) = y^2
Putting it all together, we have:
4xy^2
So, the simplified expression is:
4xy^2
3) (2 + i) / (1 - 3i)
To simplify this complex fraction, we need to remove the complex number in the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.
The conjugate of a complex number a + bi is a - bi.
In this case, the denominator is 1 - 3i, so the conjugate is 1 + 3i.
Multiplying the numerator and denominator by the conjugate, we get:
[(2 + i) * (1 + 3i)] / [(1 - 3i) * (1 + 3i)]
Now, let's simplify each multiplication term within the brackets:
(2 + i) * (1 + 3i) = 2 + 6i + i + 3i^2
Simplifying further:
2 + 6i + i + 3(-1)
Now combine like terms:
2 + 7i - 3
Resulting in:
-1 + 7i
So, the simplified expression is:
-1 + 7i