Math 225 Fall 2023 PRACTICE Final Exam QUESTIONS AND ANSWERS APPLIED LINEAR ALGEBRA

Math 225 Fall 2023 PRACTICE Final Exam QUESTIONS AND ANSWERS APPLIED LINEAR ALGEBRA 25/25 that's 100% 1 Write the expression as a single power of b. RETAKE lOMoARcPSD| RATIONALE lOMoARcPSD| Start by simplifying the terms in the parentheses. Using the Quotient Property of Exponents, divide the two terms that have the same base by subtracting the exponents, and . When subtracting fractions with a common denominator, subtract across the numerators and leave the denominator the same. minus equals . Next, apply the Power of Property of Exponents to multiply the two exponents and write b as a single power. times equals . Lastly, rewrite the fraction in its simplest form. simplifies to , so the expression simplifies to . CONCEPT Properties of Fractional and Negative Exponents 2 Write the following expression as a single complex number. lOMoARcPSD| RATIONALE When dividing complex numbers, we must clear imaginary numbers from the denominator. To do so, we can use the complex conjugate of the denominator to create a second fraction. The complex conjugate of 4 – 3i is 4 + 3i. In this second fraction, the complex conjugate 4 – 3i is the numerator and the denominator. Next, multiply across the numerators and denominators using FOIL. In both the numerator and denominator, we multiplied the first, outside, inside and last terms together. Now, we can evaluate each multiplication. Once we have evaluated the multiplication, we can combine any like terms. In the numerator, we can combine 3i and 4i to get 7i In the denominator, 12i and -12i cancel each other out. The next step is to simplify the terms that contain the imaginary unit squared. Recall that the is equivalent to -1, which can be substituted in our expression. When each in the numerator and denominator is replaced with -1, we can evaluate any multiplication. In the numerator, 3 times -1 equals -3 and in the denominator, -9 times -1 equals 9. Next, we can combine any like terms. In the numerator, 4 minus 3 equals 1. In the denominator, 16 plus 9 is equal to 25. Now we can separate into two fractions. The expression can be written as a single complex number, . CONCEPT Divide Complex Numbers 3 Consider the following table. lOMoARcPSD| x