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Evaluate j8, -1/j7, and 4/(2j13) where j is the imaginary unit.

Mathematics

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Answers

3 parts
a

1\text{1}

b

-j\text{-j}

c

-2j\text{-2j}

Step-by-step solution

12 steps

Assuming jj represents the imaginary unit, where j2=−1j^2 = -1.

Part a· 3 steps

Evaluate j8j^8

  1. Rewrite j8j^8 using j2j^2.

    j8=(j2)4j^8 = (j^2)^4

  2. Substitute j2=−1j^2 = -1.

    (j2)4=(−1)4(j^2)^4 = (-1)^4

  3. Calculate the power.

    (−1)4=1(-1)^4 = 1

    The value of j8j^8 is 1\boxed{\text{1}}.

Part b

-j\text{-j}

Evaluate −1j7-\frac{1}{j^7}

  1. Simplify the denominator j7j^7.

    j7=j4⋅j3=(j2)2⋅j2⋅j=(1)⋅(−1)⋅j=−jj^7 = j^4 \cdot j^3 = (j^2)^2 \cdot j^2 \cdot j = (1) \cdot (-1) \cdot j = -j

    Alternatively, j7=j4⋅j3=1⋅j3=j3=j2⋅j=−1⋅j=−jj^7 = j^4 \cdot j^3 = 1 \cdot j^3 = j^3 = j^2 \cdot j = -1 \cdot j = -j.

  2. Substitute the simplified denominator into the expression.

    −1j7=−1−j=1j-\frac{1}{j^7} = -\frac{1}{-j} = \frac{1}{j}

  3. Rationalize the denominator by multiplying the numerator and denominator by jj.

    1j=1⋅jj⋅j=jj2\frac{1}{j} = \frac{1 \cdot j}{j \cdot j} = \frac{j}{j^2}

  4. Substitute j2=−1j^2 = -1.

    jj2=j−1=−j\frac{j}{j^2} = \frac{j}{-1} = -j

    The value of −1j7-\frac{1}{j^7} is -j\boxed{\text{-j}}.

Part c

-2j\text{-2j}

Evaluate 42j13\frac{4}{2j^{13}}

  1. Simplify the numerical part of the fraction.

    42j13=2j13\frac{4}{2j^{13}} = \frac{2}{j^{13}}

  2. Simplify the denominator j13j^{13}.

    j13=j12⋅j=(j4)3⋅j=(1)3⋅j=1⋅j=jj^{13} = j^{12} \cdot j = (j^4)^3 \cdot j = (1)^3 \cdot j = 1 \cdot j = j

  3. Substitute the simplified denominator into the expression.

    2j13=2j\frac{2}{j^{13}} = \frac{2}{j}

  4. Rationalize the denominator by multiplying the numerator and denominator by jj.

    2j=2⋅jj⋅j=2jj2\frac{2}{j} = \frac{2 \cdot j}{j \cdot j} = \frac{2j}{j^2}

  5. Substitute j2=−1j^2 = -1.

    2jj2=2j−1=−2j\frac{2j}{j^2} = \frac{2j}{-1} = -2j

    The value of 42j13\frac{4}{2j^{13}} is -2j\boxed{\text{-2j}}.

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