University:
Stanford University
Course:
MATH 20 | Calculus
Academic year:

2023
Views:

21

Pages:

4
Author:

MathPaladin

Questions and Answers Sheet 1 Exponents and Radicals Question #1 Use the Laws of Exponents to rewrite and simplify the following expression. Be sure to eliminate any negative exponents. (3𝑥 −4 )−2 𝑥 −3 Answer: Given that: (3𝑥 −4 )−2 𝑥 −3 3 = ( 4) = = = = 𝑥 1 3 2 −2 ⋅ 𝑥 −3 ∴ 𝑥 −9 = 1 𝑥9 𝑥 −3 ( 4) 𝑥 1 32 𝑥8 𝑥8 ⋅ 𝑥 −3 ⋅ 𝑥 −3 9 𝑥 8−3 9 ∴ (3𝑥 −4 )−2 ⋅ 𝑥 −3 = 𝑥5 9 Question #2 Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents. 1 6𝑥−6 = 7776 Answer: We know that 68 = 7776 ⇒ 6−5 = 1 7776 If bases are same then power must be equal. Given equation is 1 6𝑥−6 = 7776 Using equation we can write as 1 6𝑥−6 = ⇒ 6𝑥−6 = 6−5 7776 As bases are same powers must be equal so ⇒ 𝑥 − 6 = −5 ⇒𝑥=1 Hence the equation is solved and the value of x is obtained as 1. Question #3 Simplify the expression and eliminate any negative exponents(s). Assume that all letters denote positive numbers. 6 3 √𝑦 5 √𝑦 2 Answer: 6 3 The given expression is, √𝑦 5 √𝑦 2 Simplify the expression using the laws of exponents as follows. 6 1 3 1 √𝑦 5 √𝑦 2 = (𝑦 5 )6 (𝑦 2 )3 5 2 = 𝑦 6𝑦 3 5 2 = 𝑦 6+3 =𝑦 5+4 6 9 = 𝑦6 3 = 𝑦2 3 Thus, the simplified answer is 𝑦 2 Question #4 Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents. 1 27−𝑥 = 4 Answer: Given: 1 27−𝑥 = 4 Calculation: 1 27−𝑥 = 27−𝑥 = 4 1 22 27−𝑥 = 2−2 equating exponent 7-x=-2 7+2=x x=9 The solution set is 9. Question #5 Simplify, please (4𝑥 3 )2 Answer: (4𝑥 3 )2 = 42 (𝑥 3 )2 = 16𝑥 6 Question #6 How do you simplify √18 Answer: √18 = √3 × 3 × 2 = √9 × 2 = 3√2 Question #7 Simplify, please √34. Answer: You can't really simplify √34. But, √34 = 5.83 Question #8 Simplify √128 + √32 Answer: √128 + √32 √(4 × 4) × (2 × 2) × 2 + √(4 × 4) × 2 4 × 2√2 + 4√2 8√2 + 4√2 = 12√2 Question #9 2 5 9 4 How do you use the laws of exponents to simplify (3 ) Answer: When a power is to the power of something, then we multiply the two exponents: 5 2  34 9 And simplify.. 10 336 Hope this helps! Question #10 Write in simplest radical form: a) 2√200 − √98 b) √32𝑥 3 − 𝑥√8𝑥 + 2√25 3 2 c) √ 2 3𝑥 𝑦 Answer: a) 2√200 − √98 ⇒ 2 × 10√2 − 7√2 ⇒ 13√2 Question #11 How do you write 7 × 7 × 7 × 7 × 7 in exponential form? Answer: An exponent is the same as an index and tells us what a power is made up of. 7 × 7 × 7 × 7 × 7 = 75 The powers of 7 are:7; 49;343; 2,401; 16,807. They are written with exponents as: 71 ; 72 ; 73, 74, 75 Question #12 Simplify 8−1 Answer: The negative sign tells you to take the reciprocal of 8 which is Question #13 1 Write in radical form: 2𝑥 −2 Answer: 𝑥 −𝑎 = 1 𝑥𝑎 1 Hence, we have 2𝑥 −2 = 1 2 1 𝑥2 𝑛 It is also know that 𝑥 𝑛 = √𝑥 2 2 1 = 𝑥2 √𝑥 1 8