Question 1 Exponential and Logarithmic Equations Solve the equation. Find the exact solution if possible, otherwise, use a calculator to approximate to two decimals. 𝑥 2 𝑒 2 𝑥 + 2 𝑥 𝑒 2 𝑥 = 8 𝑒 2 𝑥 x 2 e 2x +2xe 2x =8e 2x Answer 1 Step 1 For solving Exponential Equations: Isolate the exponential expression on one side of the equation. Take the logarithm of each side, then use the Laws of Logarithms to "bring down exponent". Solve for the variable. Step 2 Given, 𝑥 2 𝑒 2 𝑥 + 2 𝑥 𝑒 2 𝑥 = 8 𝑒 2 𝑥 x 2 e 2x +2xe 2x =8e 2x Add − 8 𝑒 2 𝑥 −8e 2x on both sides, 𝑥 2 𝑒 2 𝑥 + 2 𝑥 𝑒 2 𝑥 − 8 𝑒 2 𝑥 = 0 x 2 e 2x +2xe 2x −8e 2x =0 Factor 𝑒 2 𝑥 e 2x from each term, 𝑒 2 𝑥 ( 𝑥 2 + 2 𝑥 − 8 ) = 0 e 2x (x 2 +2x−8)=0 Divide by 𝑒 2 𝑥 e 2x on both sides, 𝑥 2 + 2 𝑥 − 8 = 0 x 2 +2x−8=0 Factor 𝑥 2 + 2 𝑥 − 8 x 2 +2x−8 as ( 𝑥 + 4 ) ( 𝑥 − 2 ) (x+4)(x−2) ( 𝑥 + 4 ) ( 𝑥 − 2 ) = 0 (x+4)(x−2)=0 By the zero factor property, 𝑥 + 4 = 0 and 𝑥 − 2 = 0 x+4=0 and x−2=0 Taken, 𝑥 + 4 = 0 x+4=0 Add − 4 −4 on both sides, 𝑥 = − 4 x=−4 Now taken, 𝑥 − 2 = 0 x−2=0 Add 2 on both sides, 𝑥 = 2 x=2 Thus, the exact solution is 𝑥 = { − 4 , 2 } x={−4,2}. Final statement: Therefore, the exact solution is 𝑥 = { − 4 , 2 } x={−4,2} Question 2 To find: The exact circumference of the circle with diameter 20 inches and approximate the circumference. Answer 1 Definition: Convert decimal fractions to integers. The decimals point in the product is placed so that the number of decimal places in the product is equal to the sum of the number decimal places in the factors. Radius of the circle is half the diameter of the circle. Circumference of the circle is 2 𝜋 𝑟 2πr The value of 𝜋 π is 3.14 Calculation: Radius, 𝑟 = 10 r=10 inches Circumference of the circle is 2 𝜋 𝑟 2πr Change 𝑟 = 10 r=10 and 𝜋 = 3.14 π=3.14 2 ( 3.14 ) ( 10 ) 2(3.14)(10) 20 · 3.14 Based on the definition, Multiply the decimals as through they are whole numbers. 20 ⋅ 314 = 6280 20⋅314=6280 The decimals point in the product is placed so that the number of decimal places in the product is equal to the sum of the number decimal places in the factors. The number decimal places in the factors is 2. So, place decimal point from two number from right. 62.80 On rounding off, 62.80 is taken as 63 Answer: Circumference of the circle is 63 meters Question 3 Convert 3/8 into a decimal and percent. Answer 1 3 8 = 0.375 , - is a decimal form. 8 3 =0.375, - is a decimal form. 3 8 = 37.5 % is a percentage form. 8 3 =37.5% is a percentage form. Answer 2 3 8 in decimal form is 0.375 and 37.5 in percentage form. 8 3 in decimal form is 0.375 and 37.5 in percentage form. Question 4 What is decimal notation? Answer 1 Decimal notation is the expression of numbers through powers of 10. Answer 2 Decimal notation is the representation of a fraction using the base 10 and consisting of a decimal point. Answer 3 A representation of a fraction or other real number using the base ten and consisting of any of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and a decimal point, where each digit to the left of the decimal point indicates a multiple of a positive power of ten, while each digit to the right indicates a multiple of a negative power of ten. Question 5 What is 7/8 as a decimal rounded to the nearest hundredth? Answer 1 Hundredth is the reciprocal of 100; 1 100 = 0.01 100 1 =0.01. In other words, you can interpret hundredth as nearest 2 decimal place. 7 8 = 0.875 = 0.88 (2.d.p) 8 7 =0.875=0.88 (2.d.p) Answer 2 First, we convert the fraction to a decimal number by dividing the numerator by the denominator: 7 8 = 0.875 8 7 =0.875 There are two parts to the decimal number above: Integer Part: 0 Fractional Part: 875 Now, we will make the Fractional Part just two digits (nearest hundredth) by using our rounding rules. In this case, Rule II applies, so 7 8 8 7 (or 0.875) rounded to the nearest hundredth in decimal form is: 0.88 Next, we will make 7/8 rounded to the nearest hundredth in fraction format. Since you can divide our decimal format answer above by 1 and keep the same value, you can make it like this: Answer 3 7 8 = 0.875 = 0.88 (2.d.p) 8 7 =0.875=0.88 (2.d.p)