If the expression (8-i)/(3-2i) is rewritten in the form a + bi, where a and b are real numbers, find the value of a.

Understand the Problem

The question asks to find the value of 'a' when the expression (8-i)/(3-2i) is rewritten in the form a + bi, where a and b are real numbers. This involves complex number arithmetic, specifically dividing complex numbers and expressing the result in the standard form.

Answer

$2$

Steps to Solve

Multiply by the conjugate

To divide complex numbers, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $3 - 2i$ is $3 + 2i$.

$$ \frac{8-i}{3-2i} = \frac{8-i}{3-2i} \cdot \frac{3+2i}{3+2i} $$

Expand the numerator and denominator

Now, we multiply out the numerator and the denominator:

Numerator: $$ (8-i)(3+2i) = 8(3) + 8(2i) - i(3) - i(2i) = 24 + 16i - 3i - 2i^2 $$ Since $i^2 = -1$, we have $$ 24 + 16i - 3i - 2(-1) = 24 + 13i + 2 = 26 + 13i $$

Denominator: $$ (3-2i)(3+2i) = 3(3) + 3(2i) - 2i(3) - 2i(2i) = 9 + 6i - 6i - 4i^2 $$ Since $i^2 = -1$, we have $$ 9 - 4(-1) = 9 + 4 = 13 $$

Simplify the expression

So, now we have: $$ \frac{26+13i}{13} $$ We can divide both the real and imaginary parts by 13:

$$ \frac{26}{13} + \frac{13i}{13} = 2 + i $$

Identify the value of a

The expression is now in the form $a + bi$, where $a = 2$ and $b = 1$. We are asked to find the value of $a$. Therefore, $a = 2$

$2$

More Information

The expression $\frac{8-i}{3-2i}$ simplifies to $2 + i$. Thus, the real part $a$ is 2 and the imaginary part $b$ is 1.

Tips

A common mistake is to forget to multiply both the numerator and the denominator by the conjugate. Another mistake is to incorrectly expand the products in the numerator or denominator, especially when dealing with the $i^2$ term. Also, students may forget that $i^2 = -1$.

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