Algebra Chapter 6 review
[6.1] simplify the expression using only positive exponents (-3f ) ⁻³
(-3f ) ⁻³ REMEMBER negative exponents mean fractions and you must distribute the exponent to each base... so (-3f ) ⁻³ = 1 or 1 -3³f ³ -27f ³
[6.1] Evaluate the expressions 3⁰ 3⁻⁴
3⁰ = 1, any value to the zero power = 1 3⁻⁴ Negative exponents means fraction 3⁻⁴ = 1/3⁴ = 1/91
[6.5] Solve the equation 6⁴ⁿ⁻⁵ = 6²ⁿ
6⁴ⁿ⁻⁵ = 6²ⁿ Since the bases are the same we can cancel them out and get 4n - 5 = 2n adding -2n & +5 to both sides we get 2n = 5 dividing both sides by 2 we get n = 5/2 check 4(5/2) - 5 = 2(5/2) = 20/2 - 5 = 10/2 = 10 - 5 = 5 CORRECT
[6.6] Write the next 3 terms of the geometric sequence 7, 21, 63, 189, ___, ___, ___
7, 21, 63, 189, ___, ___, ___ since the problem tells us it is a geometric sequence we can divide the following number by its previous number 21/7 = 3, this tells us the common ratio is 3 so we just have to multiply a value by 3 to get the next value 189 x 3 = 567 x 3 = 1,701 x 3 = 5,103 so these are the next values
[6.7] Write a recursive rule for the sequence 2, 4, 6, 10, 16, 26, Then write the next two terms of the sequence
NOTICE each term is the sum of the 2 preceding terms. so aₙ = aₙ-₁ + aₙ-₂ 2, 4, 6, 10, 16, 26, {26+ 16=} 42, {42 + 26 =} 68
[6.2] Evaluate the expression 32²/⁵
REMEMBER A fractional exponent gives power over root so 32²/⁵ is the fifth root of 32 squared. 32 = (2)(2)(2)(2)(2) so the fifth root of 32 is 2 then 2 squared is 4 so 32²/⁵ = 4
[6.4] Write a function that represents the balance after n years , after 10 years n = 10 $3,000 deposit that earns 6% annual interest compounded quarterly (4 times a year)
REMEMBER the formula for compound interest (annual interest partially gives several times a year) is y = P (1 + (r/n) to the power of nt y = $3,000 ( 1 + .06/4) to the power of t times 4 y = $3,000 ( 1.015) to the power of t times 4 if t = 10 the after 10 years you would have y = $3,000 ( 1.015) to the power of 40 = $5,442.06
[6.7] Write a recursive rule for the sequence n 1 2 3 4 aₙ 324 108 36 12
REMEMBER the geometric recursive rule aₙ = r times aₙ−₁ since 108/324 can be simplified to 1/3 then aₙ = (1/3) aₙ−₁
[6.3] Graph the equation f(n) = 4ⁿ - 2. Describe the domain & range
f(n) = 4ⁿ - 2 Domain {values of x} include all real numbers Range {values of y} y ≥ -2
[6.5] You deposit $1,000 in a savings account that earns 5% compounded yearly. Write an exponential equation to determine when the balance on the account will be $2,000. Solve the equation
since the formula for interest is y = a (1 + r )ⁿ where n represents the number of years we get $2,000 = $1,000 (1.05)ⁿ TO SIMPLIFY MATTERS I WILL FIRST DIVIDE BOTH SIDES BY $1,000 AND I GET 2 = (1.05)ⁿ n≈14
[6.6] Write an equation for the nth term of the geometric sequence n 1 2 3 4 aₙ 11 44 176 704 what is the 9th term of the sequence
the formula for the nth term of a geometric sequence is aₙ = a₁rⁿ⁻¹ To find r {common ratio} we divide a term by its preceding term so r = 44/11 or 4 the equation is aₙ = 11(4)ⁿ⁻¹ a₉ = 11(4)⁸ = 11 (65,536) = 720,896
[6.3] Evaluate the function for the given value of x y = 3ⁿ n = 5
y = 3ⁿ n = 5 = y = 3⁵ = 243
[6.4] does this function y = 4(0.95)ⁿ represent exponential growth or decay. Identify the percent of change.
y = 4(0.95)ⁿ is decay since the value inside the parentheses is < 1. it is 0.05 less than 1 since 1- 0.95 = 0.05 this translates into 5% since we have to multiply a decimal by 100 to get the percent
[6.2] ⁴√625
⁴√625 = (5)(5)(5)(5) = 625 so ⁴√625 =5