Question

simplify using only positive exponents:
(2t)^-6

Answer 1

can you help me bot

Answer 2

Certainly!

To simplify (2t)^-6 using only positive exponents, we can use the rule for negative exponents:

a^-n = 1 / a^n

In this case, a=2t and n=6:

(2t)^-6 = 1 / (2t)^6

Now we can expand and simplify the denominator using the rule for exponents:

(2t)^6 = 2^6 * t^6 = 64t^6

So we have:

(2t)^-6 = 1 / (2t)^6 = 1 / (64t^6)

And that's the simplified form with only positive exponents!

Answer 3

simplify using only positive exponents:
(w^-2 J^-4)^-3(J^7J^3)

Answer 4

Sure! Here's how we can simplify this expression using only positive exponents:

First, we'll simplify the expression inside the parentheses and combine like terms:

J^7J^3 = J^(7+3) = J^10

Now we can rewrite the expression with the negative exponent as a fraction with a positive exponent:

(w^-2 J^-4)^-3 = (1/w^2J^4)^3 = (1/w^6J^12)

Finally, we'll substitute this into the original expression, and simplify further:

(w^-2 J^-4)^-3(J^7J^3) = (1/w^6J^12) * J^10 = J^(10-12) / w^6 = J^-2 / w^6

So, the simplified expression with only positive exponents is J^-2 / w^6.

Answer 5

simplify using only positive exponents:
a^2b^-7c^4/a^5b^3C^-2

Answer 6

Sure! Here's how we can simplify the expression using only positive exponents:

First, we can simplify the individual factors inside the fraction by applying the rules of exponents:

a^2b^-7c^4 = a^2/c^4b^7

a^5b^3c^-2 = b^3a^5/c^2

Now we can substitute these into the fraction and simplify further:

a^2b^-7c^4/a^5b^3C^-2 = (a^2/c^4b^7) / (b^3a^5/c^2)

We can simplify this further by dividing the coefficients and variables separately:

(a^2/c^4b^7) / (b^3a^5/c^2) = (a^2/c^4 * c^2 / b^7b^3 * 1/a^5)

= (a^2a^-5) / (b^10c^2)

= a^(-3) / (b^10c^2)

So, the simplified expression with only positive exponents is a^(-3) / b^10c^2.

Answer 7

Evaluate the expression for m = 2, t = −3, and z = 0. z−t(mt)z

Answer 8

When m=2, t=-3, and z=0, we can plug in these values to evaluate the expression:

z - t(mt)z

= 0 - (-3) * (2*(-3)*0) * 0

= 0 - (-3) * 0 * 0

= 0

So the value of the expression when m=2, t=-3, and z=0 is 0.

Answer 9

Use scientific notation to rewrite the number: a. 0.0002603 in scientific notation
b. 5.38 × 102 in standard notation

Answer 10

(a) To write 0.0002603 in scientific notation, we need to move the decimal point to the right until there is only one non-zero digit to the left of the decimal. We count the number of places we move the decimal point and use that as the exponent of 10.

0.0002603 = 2.603 x 0.0001

We moved the decimal point four places to the right, so the exponent of 10 is -4. Therefore, 0.0002603 in scientific notation is:

2.603 x 10^-4

(b) To convert 5.38 x 10^2 to standard notation, we just write out the digits and move the decimal point to the right or left depending on the sign of the exponent.

5.38 x 10^2 = 538

We moved the decimal point two places to the right, which is the same as multiplying by 100, so 5.38 x 10^2 in standard notation is:

538

Answer 11

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.

Answer 12

We can use the formula:

time = distance / speed

to find how long it takes for sound to travel a distance of 7.2 x 10^2 km, given that the speed of sound is 1.2 x 10^3 km/h.

Substituting the given values in the formula, we get:

time = (7.2 x 10^2 km) / (1.2 x 10^3 km/h)

time = 0.6 h

We can convert this to minutes by multiplying by 60 because there are 60 minutes in an hour:

time = 0.6 h * 60 min/h

time = 36 min

Therefore, it takes 36 minutes for sound to travel a distance of 7.2 x 10^2 km at a speed of 1.2 x 10^3 km/h.

Answer 13

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?
x
3
y =⎛ ⎞
⎜ ⎟
4
⎝ ⎠

Answer 14

To evaluate the function y = (x/4) over the domain {-1, 0, 1, 2}, we substitute each of the given values of x and simplify:

When x = -1, y = (-1/4) = -0.25
When x = 0, y = (0/4) = 0
When x = 1, y = (1/4) = 0.25
When x = 2, y = (2/4) = 0.5

As the values of the domain increase from -1 to 2, we can see that the values of the function gradually increase from -0.25 to 0.5. Therefore, we can say that the values of the function increase as the values of the domain increase.

Answer 15

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.

Answer 16

To solve this problem, we can use the exponential growth formula:

A = P(1 + r)^t

where A is the ending amount, P is the initial amount, r is the growth rate, and t is the time in years.

Since the investment doubles every 12 years, we can find the yearly growth rate by finding the value that multiplied by itself three times equals 2 (because 36 years = 3 x 12 years):

2^(1/3) ≈ 1.2599

So the growth rate, r, is 1.2599 - 1 = 0.2599 or approximately 26%.

Now we can write the exponential equation for the investment after 36 years:

A = 5000(1 + 0.2599)^3

A ≈ $19,882.69

Therefore, the investment is worth approximately $19,882.69 after 36 years.

To find the value of the investment after 48 years, we can plug in t = 4 (because 48 years = 4 x 12 years) into the same formula:

A = 5000(1 + 0.2599)^4

A ≈ $31,736.23

Therefore, the investment is worth approximately $31,736.23 after 48 years.

Answer 17

Does the function represent exponential growth or exponential decay? Identify the growth or decay factor.
y = 9 ∙ (1/2)^x

Answer 18

The function y = 9 ∙ (1/2)^x represents exponential decay.

We can see that the base of the exponential term (1/2) is less than 1 and this indicates decay.

The decay factor is 1/2 since that is the value that the base is raised to. This means that the function decreases by a factor of 1/2 each time x increases by 1.

Answer 19

You deposit $520 in an account with 4% interest compounded monthly. What is the balance in the account after 5 years?

Answer 20

We can use the formula for compound interest:

A = P(1 + r/n)^(nt)

where A is the ending balance, P is the principal (initial deposit), r is the annual interest rate as a decimal, n is the number of times the interest is compounded per year, and t is the time in years.

In this case, P = $520, r = 0.04, n = 12 (compounded monthly), and t = 5.

Substituting these values into the formula, we get:

A = $520(1 + 0.04/12)^(12*5)

A ≈ $674.94

So the balance in the account after 5 years is approximately $674.94.