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.

$3 i (2 + 5 i) + (6 - 7 i) - (9 + i)$

Answer

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Solution:

Step 1: Eliminate the parentheses

Eliminate the parentheses in the expression: $3 i (2 + 5 i) + 6 - 7 i - (9 + i)$ .

Step 2: Expand and simplify the expression

Step 2.1: Distribute the imaginary unit

Distribute $3 i$ across $(2 + 5 i)$ and expand: $3 i \cdot 2 + 3 i (5 i) + 6 - 7 i - (9 + i)$ .

Step 2.2: Perform multiplication of real numbers

Multiply $2$ by $3 i$ : $6 i + 3 i (5 i) + 6 - 7 i - (9 + i)$ .

Step 2.3: Multiply the complex terms

Step 2.3.1: Multiply real numbers

Multiply $5$ by $3 i$ : $6 i + 15 i i + 6 - 7 i - (9 + i)$ .

Step 2.3.2: Apply the identity of $i$

Recognize that $i$ raised to the power of $1$ is $i$ : $6 i + 15 (i^{1} i) + 6 - 7 i - (9 + i)$ .

Step 2.3.3: Apply the identity of $i$ again

Recognize that $i$ raised to the power of $1$ is $i$ : $6 i + 15 (i^{1} i^{1}) + 6 - 7 i - (9 + i)$ .

Step 2.3.4: Combine exponents using the power rule

Apply the power rule $a^{m} a^{n} = a^{m + n}$ : $6 i + 15 i^{1 + 1} + 6 - 7 i - (9 + i)$ .

Step 2.3.5: Sum the exponents

Add the exponents $1$ and $1$ : $6 i + 15 i^{2} + 6 - 7 i - (9 + i)$ .

Step 2.4: Simplify the expression further

Step 2.4.1: Replace $i^{2}$ with $- 1$

Substitute $i^{2}$ with $- 1$ : $6 i + 15 \cdot - 1 + 6 - 7 i - (9 + i)$ .

Step 2.4.2: Multiply real numbers

Multiply $15$ by $- 1$ : $6 i - 15 + 6 - 7 i - (9 + i)$ .

Step 2.5: Distribute the negative sign

Distribute the negative sign across $(9 + i)$ : $6 i - 15 + 6 - 7 i - 1 \cdot 9 - i$ .

Step 2.6: Perform multiplication

Multiply $- 1$ by $9$ : $6 i - 15 + 6 - 7 i - 9 - i$ .

Step 3: Combine like terms

Step 3.1: Combine the imaginary terms

Combine $6 i$ and $- 7 i$ : $- 15 + 6 - i - 9 - i$ .

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 $- i$ : $- 18 - 2 i$ .

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, $a (b + c) = a b + a c$ .
Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power. For instance, $2 x + 3 x = 5 x$ .
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.