How do I do these?
1. Simplify using only positive exponents:
(2t)⁻⁶
2. Simplify using only positive exponents:
(w⁻²j⁻⁴)⁻³(j⁷j³)
3. Simplify using only positive exponents:
a²b⁻⁷c⁴
----------
a⁵b³c⁻²
4. Evaluate the expression for m = 2, t = -3, and z = 0.
z⁻ᵗ(mᵗ)ᶻ
5. Use scientific notation to rewrite the number:
a. 0.0002603 in scientific notation
b. 5.38 × 102 in standard notation
6. The speed of sound is approximately 1.2 × 10³ km/h.
How long does it take for sound to travel 7.2 × 10²
km? Write your answer in minutes.
7. Evaluate the function below over the domain {-1, 0,
1, 2}. As the values of the domain increase, do the
values of the function increase or decrease?
y = (3/4)ˣ
8. Suppose an investment of $5,000 doubles every 12
years. How much is the investment worth after 36
years? After 48 years?
Write and solve an exponential equation.
9. Does the function represent exponential growth or
exponential decay? Identify the growth or decay
factor.
= 9 ∙ (1/2)ˣ
10. You deposit $520 in an account with 4% interest
compounded monthly. What is the balance in the
account after 5 years?
I understand, I'm asking how to do them though, not to be done for me. :)
for example
2. Simplify using only positive exponents:
(w⁻²j⁻⁴)⁻³(j⁷j³)
(w^-2 j^-4)^-3 = w^6 j^12
multiply by j^10
w^6 j^22
rules
a^-x = 1/a^x
(a^x)^y = a^(xy)
a^x a^y = a(x+y)
example 10
10. You deposit $520 in an account with 4% interest
compounded monthly. What is the balance in the
account after 5 years?
4% per year is .04/12 = .0033333 per month
every month multiply by
1.0033333
do that for 60 months which is 5 years
520 * 1.0033333^60
= 520 * 1.221
= 634.92
9. Does the function represent exponential growth or
exponential decay? Identify the growth or decay
factor.
= 9 ∙ (1/2)ˣ
(1/2)^1 = 1/2
(1/2) ^2 = 1/2^2 = 1/4
(1/2)^3 = 1/2^3 = 1/8 is this getting bigger or smaller ?
(1/2)^x = 1/2^x
(1/2)^x = e^(kx)
x ln .5 = k x
k = ln .5 = -.6931
so in standard exponential notation this is
9 e^-.6931 x
see
https://www.mathsisfun.com/algebra/exponential-growth.html
1. To simplify the expression (2t)⁻⁶ using only positive exponents, we can rewrite it as 1/(2t)⁶. This means we need to move the base (2t) from the denominator to the numerator. To do this, we can change the sign of the exponent from negative to positive. So the simplified expression becomes 1/(2t)⁶ = 1/(2⁶t⁶) = 1/64t⁶.
2. To simplify the expression (w⁻²j⁻⁴)⁻³(j⁷j³) using only positive exponents, we can apply the negative exponents rule. This rule states that when a term with a negative exponent is raised to a positive exponent, we can move it from the numerator to the denominator (or vice versa) by changing the sign of the exponent. So the expression becomes (1/(w²j⁴))³(j⁷j³). Simplifying further, we have 1/(w⁶j¹²) * j¹⁰ = j¹⁰/w⁶j¹² = j¹⁰/j¹²w⁶ = 1/j²w⁶.
3. To simplify the expression a²b⁻⁷c⁴ / a⁵b³c⁻² using only positive exponents, we can apply the quotient rule. This rule states that when dividing two terms with the same base, we can subtract the exponents. So the expression becomes a²/a⁵ * b⁻⁷/b³ * c⁴/c⁻² = a²⁻⁵ * b⁻⁷⁻³ * c⁴⁺² = a⁻³ * b⁻¹⁰ * c⁶.
4. To evaluate the expression z⁻ᵗ(mᵗ)ᶻ for m = 2, t = -3, and z = 0, we substitute these values into the expression. However, note that having zero as an exponent will result in the expression being undefined. Therefore, we cannot evaluate this expression.
5.
a. To rewrite the number 0.0002603 in scientific notation, we move the decimal point to the right until there is only one nonzero digit to the left. In this case, we need to move the decimal point 4 places to the right, which gives us 2.603. The original number is less than 1, so the exponent in scientific notation will be negative. Therefore, 0.0002603 in scientific notation is 2.603 × 10⁻⁴.
b. To convert the number 5.38 × 10² from scientific notation to standard notation, we move the decimal point to the right 2 places, since the exponent is positive. This gives us 538.
6. To find out how long it takes for sound to travel 7.2 × 10² km, given the speed of sound of 1.2 × 10³ km/h, we need to divide the distance by the speed. The formula is time = distance / speed. So, time = (7.2 × 10²) / (1.2 × 10³) = 6 minutes.
7. To evaluate the function y = (3/4)ˣ over the domain {-1, 0, 1, 2}, we substitute each value in the domain into the function. For x = -1, y = (3/4)⁻¹ = 4/3. For x = 0, y = (3/4)⁰ = 1. For x = 1, y = (3/4)¹ = 3/4. For x = 2, y = (3/4)² = 9/16. As the values of the domain increase, the values of the function increase as well.
8. If an investment of $5,000 doubles every 12 years, we can write it as an exponential equation in the form A = P(1 + r)^t, where A is the amount after t years, P is the initial investment, r is the growth rate, and t is the time in years. In this case, the growth rate is 100% because the investment doubles. So the equation becomes A = 5000(1 + 1)^t = 5000(2)^t.
To find the worth of the investment after 36 years, we substitute t = 36 into the equation: A = 5000(2)^36. By evaluating the expression, we find that the investment is worth $144,000.
Similarly, to find the worth of the investment after 48 years, we substitute t = 48: A = 5000(2)^48. Evaluating the expression, we find that the investment is worth $384,000.
9. The function y = 9 x (1/2)^x represents exponential decay because the base (1/2) is less than 1. The decay factor is 1/2.
10. To calculate the balance in the account after 5 years with a $520 deposit and 4% interest compounded monthly, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where A is the final amount, P is the principal amount (deposit), r is the annual interest rate (as a decimal), n is the number of times interest is compounded per year, and t is the number of years.
Substituting the given values, we have A = 520(1 + 0.04/12)^(12*5).
Evaluating the expression, we find that the balance in the account after 5 years is approximately $604.02.