Complex Numbers Practice

Factor out square root of negative one as i, then simplify the remaining positive radical.

Write \(\sqrt{-12}=i\sqrt{12}\), then extract the largest perfect-square factor. The simplified result is \(2i\sqrt3\).

Factor out square root of negative one as i, then simplify the remaining positive radical.

Write \(\sqrt{-27}=i\sqrt{27}\), then extract the largest perfect-square factor. The simplified result is \(3i\sqrt3\).

Factor out square root of negative one as i, then simplify the remaining positive radical.

Write \(\sqrt{-32}=i\sqrt{32}\), then extract the largest perfect-square factor. The simplified result is \(4i\sqrt2\).

Factor out square root of negative one as i, then simplify the remaining positive radical.

Write \(\sqrt{-75}=i\sqrt{75}\), then extract the largest perfect-square factor. The simplified result is \(5i\sqrt3\).

Factor out square root of negative one as i, then simplify the remaining positive radical.

Write \(\sqrt{-98}=i\sqrt{98}\), then extract the largest perfect-square factor. The simplified result is \(7i\sqrt2\).

Reduce the exponent modulo 4 and use the cycle 1, i, -1, -i.

The powers of \(i\) repeat every four. Since \(17\) leaves remainder \(1\) modulo \(4\), \(i^{17}=i\).

Reduce the exponent modulo 4 and use the cycle 1, i, -1, -i.

The powers of \(i\) repeat every four. Since \(26\) leaves remainder \(2\) modulo \(4\), \(i^{26}=-1\).

Reduce the exponent modulo 4 and use the cycle 1, i, -1, -i.

The powers of \(i\) repeat every four. Since \(35\) leaves remainder \(3\) modulo \(4\), \(i^{35}=-i\).

Reduce the exponent modulo 4 and use the cycle 1, i, -1, -i.

The powers of \(i\) repeat every four. Since \(44\) leaves remainder \(0\) modulo \(4\), \(i^{44}=1\).

Reduce the exponent modulo 4 and use the cycle 1, i, -1, -i.

The powers of \(i\) repeat every four. Since \(59\) leaves remainder \(3\) modulo \(4\), \(i^{59}=-i\).

Match the expression to a+bi and distinguish b from the complete term bi.

In standard form \(a+bi\), the real part is \(a\), the imaginary coefficient is \(b\), and the imaginary term is \(bi\). Therefore the requested real part is \(7\).

Match the expression to a+bi and distinguish b from the complete term bi.

In standard form \(a+bi\), the real part is \(a\), the imaginary coefficient is \(b\), and the imaginary term is \(bi\). Therefore the requested imaginary coefficient is \(9\).

Match the expression to a+bi and distinguish b from the complete term bi.

In standard form \(a+bi\), the real part is \(a\), the imaginary coefficient is \(b\), and the imaginary term is \(bi\). Therefore the requested imaginary coefficient is \(-1\).

Match the expression to a+bi and distinguish b from the complete term bi.

In standard form \(a+bi\), the real part is \(a\), the imaginary coefficient is \(b\), and the imaginary term is \(bi\). Therefore the requested imaginary term is \(-5i\).

Match the expression to a+bi and distinguish b from the complete term bi.

In standard form \(a+bi\), the real part is \(a\), the imaginary coefficient is \(b\), and the imaginary term is \(bi\). Therefore the requested real part is \(11\).

Line up real parts with real parts and coefficients of i with coefficients of i.

Add real parts and imaginary coefficients separately: \(3+(5)=8\) and \(4+(-2)=2\). The result is \(8+2i\).

Line up real parts with real parts and coefficients of i with coefficients of i.

Add real parts and imaginary coefficients separately: \(-6+(2)=-4\) and \(3+(7)=10\). The result is \(-4+10i\).

Line up real parts with real parts and coefficients of i with coefficients of i.

Add real parts and imaginary coefficients separately: \(8+(-3)=5\) and \(-5+(-4)=-9\). The result is \(5-9i\).

Line up real parts with real parts and coefficients of i with coefficients of i.

Add real parts and imaginary coefficients separately: \(1+(7)=8\) and \(9+(1)=10\). The result is \(8+10i\).

Line up real parts with real parts and coefficients of i with coefficients of i.

Add real parts and imaginary coefficients separately: \(-4+(-5)=-9\) and \(-6+(8)=2\). The result is \(-9+2i\).

Place parentheses around the subtracted complex number before combining like parts.

Distribute subtraction across both parts: real part \(7-(3)=4\), imaginary coefficient \(2-(5)=-3\). Thus the result is \(4-3i\).

Place parentheses around the subtracted complex number before combining like parts.

Distribute subtraction across both parts: real part \(-1-(4)=-5\), imaginary coefficient \(6-(-2)=8\). Thus the result is \(-5+8i\).

Place parentheses around the subtracted complex number before combining like parts.

Distribute subtraction across both parts: real part \(9-(-2)=11\), imaginary coefficient \(-3-(7)=-10\). Thus the result is \(11-10i\).

Place parentheses around the subtracted complex number before combining like parts.

Distribute subtraction across both parts: real part \(5-(6)=-1\), imaginary coefficient \(8-(1)=7\). Thus the result is \(-1+7i\).

Place parentheses around the subtracted complex number before combining like parts.

Distribute subtraction across both parts: real part \(-4-(-7)=3\), imaginary coefficient \(-5-(-2)=-3\). Thus the result is \(3-3i\).

Use (a+bi)(c+di)=(ac-bd)+(ad+bc)i and verify the sign from i squared.

FOIL and replace \(i^2\) by \(-1\). The real part is \(2(4)-3(-1)=11\), and the imaginary coefficient is \(2(-1)+3(4)=10\).

Use (a+bi)(c+di)=(ac-bd)+(ad+bc)i and verify the sign from i squared.

FOIL and replace \(i^2\) by \(-1\). The real part is \(5(1)--2(3)=11\), and the imaginary coefficient is \(5(3)+-2(1)=13\).

Use (a+bi)(c+di)=(ac-bd)+(ad+bc)i and verify the sign from i squared.

FOIL and replace \(i^2\) by \(-1\). The real part is \(-3(2)-4(2)=-14\), and the imaginary coefficient is \(-3(2)+4(2)=2\).

Use (a+bi)(c+di)=(ac-bd)+(ad+bc)i and verify the sign from i squared.

FOIL and replace \(i^2\) by \(-1\). The real part is \(1(-2)--5(3)=13\), and the imaginary coefficient is \(1(3)+-5(-2)=13\).

Use (a+bi)(c+di)=(ac-bd)+(ad+bc)i and verify the sign from i squared.

FOIL and replace \(i^2\) by \(-1\). The real part is \(6(3)-1(-4)=22\), and the imaginary coefficient is \(6(-4)+1(3)=-21\).

A complex number times its conjugate equals the sum of the squares of its two coefficients.

The conjugate is \(3-4i\). Their product is \(3^2+(4)^2=25\), a real number.

A complex number times its conjugate equals the sum of the squares of its two coefficients.

The conjugate is \(-2-5i\). Their product is \(-2^2+(5)^2=29\), a real number.

A complex number times its conjugate equals the sum of the squares of its two coefficients.

The conjugate is \(7+i\). Their product is \(7^2+(-1)^2=50\), a real number.

A complex number times its conjugate equals the sum of the squares of its two coefficients.

The conjugate is \(1-6i\). Their product is \(1^2+(6)^2=37\), a real number.

A complex number times its conjugate equals the sum of the squares of its two coefficients.

The conjugate is \(-4+3i\). Their product is \(-4^2+(-3)^2=25\), a real number.

Use the denominator's conjugate so its product becomes a real sum of squares.

Multiply numerator and denominator by the conjugate of \(2-i\). The denominator becomes \(5\), and simplifying the numerator gives \(\frac95+\frac75i\).

Use the denominator's conjugate so its product becomes a real sum of squares.

Multiply numerator and denominator by the conjugate of \(1-2i\). The denominator becomes \(5\), and simplifying the numerator gives \(-1+2i\).

Use the denominator's conjugate so its product becomes a real sum of squares.

Multiply numerator and denominator by the conjugate of \(3+i\). The denominator becomes \(10\), and simplifying the numerator gives \(2-i\).

Use the denominator's conjugate so its product becomes a real sum of squares.

Multiply numerator and denominator by the conjugate of \(1+i\). The denominator becomes \(2\), and simplifying the numerator gives \(3-i\).

Use the denominator's conjugate so its product becomes a real sum of squares.

Multiply numerator and denominator by the conjugate of \(2+i\). The denominator becomes \(5\), and simplifying the numerator gives \(\frac75+\frac95i\).

Write one equation for real parts and another for imaginary coefficients.

Equal complex numbers have matching parts. From \(a+2=8\), \(a=6\). From \(3b-1=11\), \(b=4\). Therefore \(a+b=10\).

Write one equation for real parts and another for imaginary coefficients.

Equal complex numbers have matching parts. From \(a-4=3\), \(a=7\). From \(2b+5=-1\), \(b=-3\). Therefore \(a+b=4\).

Write one equation for real parts and another for imaginary coefficients.

Equal complex numbers have matching parts. From \(a+7=-2\), \(a=-9\). From \(-1b+4=9\), \(b=-5\). Therefore \(a+b=-14\).

Write one equation for real parts and another for imaginary coefficients.

Equal complex numbers have matching parts. From \(a-3=5\), \(a=8\). From \(4b-2=10\), \(b=3\). Therefore \(a+b=11\).

Write one equation for real parts and another for imaginary coefficients.

Equal complex numbers have matching parts. From \(a+1=6\), \(a=5\). From \(-2b+3=-7\), \(b=5\). Therefore \(a+b=10\).

Reduce the work to a+bi form and check every occurrence of i squared or a conjugate sign.

Complex arithmetic follows ordinary distribution and like-term rules together with the defining identity \(i^2=-1\). The stated correction preserves those rules.

Reduce the work to a+bi form and check every occurrence of i squared or a conjugate sign.

Complex arithmetic follows ordinary distribution and like-term rules together with the defining identity \(i^2=-1\). The stated correction preserves those rules.

Reduce the work to a+bi form and check every occurrence of i squared or a conjugate sign.

Complex arithmetic follows ordinary distribution and like-term rules together with the defining identity \(i^2=-1\). The stated correction preserves those rules.

Reduce the work to a+bi form and check every occurrence of i squared or a conjugate sign.

Complex arithmetic follows ordinary distribution and like-term rules together with the defining identity \(i^2=-1\). The stated correction preserves those rules.

Reduce the work to a+bi form and check every occurrence of i squared or a conjugate sign.

Complex arithmetic follows ordinary distribution and like-term rules together with the defining identity \(i^2=-1\). The stated correction preserves those rules.