University:
Jackson College
Course:
MAT-010 | Basic Mathematics
Academic year:

2024
Views:

339

Pages:

3
Author:

Bennett Wheeler

Question 1 Solve the exponential equation 3 2 𝑥 = 8 32 x =8 by expressing each side as a power of the same base and then equating exponents. Answer 1 To determine: To solve 3 2 𝑥 = 8 32 x =8 by expressing each side as a power of the same base and then equating exponents. Calculation: We know 32 = 2 5 32=2 5 Also, 8 = 2 3 8=2 3 Plugging these values in the given equation, we get: ( 2 5 ) 𝑥 = 2 3 (2 5 ) x =2 3 If bases are the same, equating powers, we get: 5 𝑥 = 3 5x=3 𝑥 = 3 5 x= 5 3 Hence, 𝑥 = 3 5 x= 5 3 is the solution of the given equation. Question 2 Use the laws of exponents to simplify the expression ( 6 4 1 2 ) 2 (64 2 1 ) 2 . Answer 1 Using the laws of exponents to simplify the expressions: Law of exponents: ( 𝑎 𝑚 ) 𝑛 = 𝑎 𝑚 ⋅ 𝑛 (a m ) n =a m⋅n Simplifying the given expression: ( 6 4 1 2 ) 2 = 6 4 1 2 ⋅ 2 (64 2 1 ) 2 =64 2 1 ⋅2 = 6 4 1 =64 1 = 64 =64 Hence, the simplified answer is 64. Question 3 Write each root using exponents and evaluate − 81 4 4 −81 . Answer 1 To write: Root − 81 4 4 −81 using exponents and evaluate. Concept used: Exponent rule: ( 𝑏 𝑛 ) 1 𝑎 = 𝑏 𝑛 𝑎 (b n ) a 1 =b a n Calculation: The root − 81 4 4 −81 can be written as: − 81 4 = ( − 81 ) 1 4 = ( − 1 ) 1 4 ( 3 4 ) 1 4 4 −81 =(−81) 4 1 =(−1) 4 1 (3 4 ) 4 1 Thus, the root − 81 4 4 −81 using exponents can be written as 3 ( − 1 ) 1 4 3(−1) 4 1 . Question 4 Condense each expression. Answers should not contain negative exponents. Assume that every variable is a positive real number. ( 121 100 ) − 3 2 ( 100 121 ) − 2 3 Answer 1 We need to consider the expression ( 121 100 ) − 3 2 ( 100 121 ) − 2 3 and use the exponent property ( 𝑎 𝑏 ) − 𝑚 = ( 𝑏 𝑎 ) 𝑚 ( b a ) −m =( a b ) m to simplify this expression. Since 100 and 121 can be written as 1 0 2 10 2 and 1 1 2 11 2 : = ( 1 0 2 1 1 2 ) 3 2 =( 11 2 10 2 ) 2 3 Using the exponent rule ( 𝑎 𝑚 ) 𝑛 = 𝑎 𝑚 ⋅ 𝑛 (a m ) n =a m⋅n , we get: = 1 0 3 1 1 3 = 11 3 10 3 Hence, ( 121 100 ) − 3 2 = 1 0 3 1 1 3 ( 100 121 ) − 2 3 = 11 3 10 3 . Question 5 Use the Laws of Exponents to rewrite and simplify the following expression. Be sure to eliminate any negative exponents. ( 3 𝑥 − 4 ) − 2 𝑥 − 3 (3x −4 ) −2 x −3 Answer 1 Given: ( 3 𝑥 − 4 ) − 2 𝑥 − 3 (3x −4 ) −2 x −3 Simplify ( 3 𝑥 − 4 ) − 2 (3x −4 ) −2 : = ( 3 𝑥 4 ) − 2 =( x 4 3 ) −2 = 1 3 2 𝑥 4 ⋅ 2 = 3 2 1 x 4⋅2 = 1 9 𝑥 8 = 9 1 x 8 Combine with 𝑥 − 3 x −3 : = 1 9 𝑥 8 − 3 = 9 1 x 8−3 = 1 9 𝑥 5 = 9 1 x 5 Hence, the simplified expression is: 𝑥 5 9 9 x 5