Simplify 3i(2+5i)+(6-7i)-(9+i)
The problem involves complex numbers and asks for the simplification of an expression that contains both real and imaginary parts. You must perform operations of multiplication and addition/subtraction of complex numbers, taking into account the imaginary unit 'i', which has the property that i² = -1. The operations must be carried out following the standard algebraic rules for combining like terms and applying the distributive property where necessary.
$3 i \left(\right. 2 + 5 i \left.\right) + \left(\right. 6 - 7 i \left.\right) - \left(\right. 9 + i \left.\right)$
Answer
Solution:
Step 1: Eliminate the parentheses
Eliminate the parentheses in the expression: $3i(2 + 5i) + 6 - 7i - (9 + i)$.
Step 2: Expand and simplify the expression
Step 2.1: Distribute the imaginary unit
Distribute $3i$ across $(2 + 5i)$ and expand: $3i \cdot 2 + 3i(5i) + 6 - 7i - (9 + i)$.
Step 2.2: Perform multiplication of real numbers
Multiply $2$ by $3i$: $6i + 3i(5i) + 6 - 7i - (9 + i)$.
Step 2.3: Multiply the complex terms
Step 2.3.1: Multiply real numbers
Multiply $5$ by $3i$: $6i + 15ii + 6 - 7i - (9 + i)$.
Step 2.3.2: Apply the identity of $i$
Recognize that $i$ raised to the power of $1$ is $i$: $6i + 15(i^1 i) + 6 - 7i - (9 + i)$.
Step 2.3.3: Apply the identity of $i$ again
Recognize that $i$ raised to the power of $1$ is $i$: $6i + 15(i^1 i^1) + 6 - 7i - (9 + i)$.
Step 2.3.4: Combine exponents using the power rule
Apply the power rule $a^m a^n = a^{m+n}$: $6i + 15i^{1+1} + 6 - 7i - (9 + i)$.
Step 2.3.5: Sum the exponents
Add the exponents $1$ and $1$: $6i + 15i^2 + 6 - 7i - (9 + i)$.
Step 2.4: Simplify the expression further
Step 2.4.1: Replace $i^2$ with $-1$
Substitute $i^2$ with $-1$: $6i + 15 \cdot -1 + 6 - 7i - (9 + i)$.
Step 2.4.2: Multiply real numbers
Multiply $15$ by $-1$: $6i - 15 + 6 - 7i - (9 + i)$.
Step 2.5: Distribute the negative sign
Distribute the negative sign across $(9 + i)$: $6i - 15 + 6 - 7i - 1 \cdot 9 - i$.
Step 2.6: Perform multiplication
Multiply $-1$ by $9$: $6i - 15 + 6 - 7i - 9 - i$.
Step 3: Combine like terms
Step 3.1: Combine the imaginary terms
Combine $6i$ and $-7i$: $-15 + 6 - i - 9 - i$.
Step 3.2: Add and subtract real numbers
Step 3.2.1: Add $-15$ and $6$
Combine the real numbers: $-9 - i - 9 - i$.
Step 3.2.2: Subtract $9$ from $-9$
Subtract $9$ from $-9$: $-18 - i - i$.
Step 3.3: Combine the imaginary terms
Combine $-i$ and $-i$: $-18 - 2i$.
Knowledge Notes:
To solve this problem, we used several algebraic rules and properties:
Distributive Property: This property allows us to multiply a single term by each term within a parenthesis. For example, $a(b + c) = ab + ac$.
Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power. For instance, $2x + 3x = 5x$.
Imaginary Unit: The imaginary unit $i$ is defined such that $i^2 = -1$. This is used to simplify expressions involving $i$.
Exponent Rules: When multiplying like bases, we add the exponents, as in $a^m \cdot a^n = a^{m+n}$.
Simplifying Complex Numbers: Complex numbers consist of a real part and an imaginary part. When simplifying, we combine the real parts and the imaginary parts separately.
In the given problem, we applied these rules to simplify a complex expression involving the imaginary unit $i$. The process involved expanding the expression, simplifying using the properties of $i$, and combining like terms to reach the final simplified form.
