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3 years ago

Simplify (6^5/7^3)^2 A.6^7/7^10 B. 6^10/7^6 C. 6^3/7 D.12^5/14^3

Mathematics

1 answer:

ivolga24 [154]3 years ago

8 0

Answer:

answer is B

Step-by-step explanation:

all work is pictured and shown

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riadik2000 [5.3K]

Let

$S_n = \displaystyle \sum_{k=0}^n r^k = 1 + r + r^2 + \cdots + r^n$

where we assume |r| < 1. Multiplying on both sides by r gives

$r S_n = \displaystyle \sum_{k=0}^n r^{k+1} = r + r^2 + r^3 + \cdots + r^{n+1}$

and subtracting this from $S_n$ gives

$(1 - r) S_n = 1 - r^{n+1} \implies S_n = \dfrac{1 - r^{n+1}}{1 - r}$

As n → ∞, the exponential term will converge to 0, and the partial sums $S_n$ will converge to

$\displaystyle \lim_{n\to\infty} S_n = \dfrac1{1-r}$

Now, we're given

$a + ar + ar^2 + \cdots = 15 \implies 1 + r + r^2 + \cdots = \dfrac{15}a$

$a^2 + a^2r^2 + a^2r^4 + \cdots = 150 \implies 1 + r^2 + r^4 + \cdots = \dfrac{150}{a^2}$

We must have |r| < 1 since both sums converge, so

$\dfrac{15}a = \dfrac1{1-r}$

$\dfrac{150}{a^2} = \dfrac1{1-r^2}$

Solving for r by substitution, we have

$\dfrac{15}a = \dfrac1{1-r} \implies a = 15(1-r)$

$\dfrac{150}{225(1-r)^2} = \dfrac1{1-r^2}$

Recalling the difference of squares identity, we have

$\dfrac2{3(1-r)^2} = \dfrac1{(1-r)(1+r)}$

We've already confirmed r ≠ 1, so we can simplify this to

$\dfrac2{3(1-r)} = \dfrac1{1+r} \implies \dfrac{1-r}{1+r} = \dfrac23 \implies r = \dfrac15$

It follows that

$\dfrac a{1-r} = \dfrac a{1-\frac15} = 15 \implies a = 12$

and so the sum we want is

$ar^3 + ar^4 + ar^6 + \cdots = 15 - a - ar - ar^2 = \boxed{\dfrac3{25}}$

which doesn't appear to be either of the given answer choices. Are you sure there isn't a typo somewhere?

7 0

3 years ago

The slope of a hill next to a highway decreases 2 vertical feet for every 3 horizontal feet away from the highway.

Novay_Z [31]

Slope is the change in Y ( vertical) over the change in X( horizontal).

The slope would be -2 /3

You would multiply the horizontal distance by the slope:

-2/3 x 12 = -8

The answer would be D. v(h)=-2/3h, and the vertical distance is -8 feet.

8 0

3 years ago

What is 5.17490 rounded to the nearest thousandth

Annette [7]

<span>5.17490 rounded to the nearest thousandth is 5.175</span>

3 0

3 years ago

Read 2 more answers

2 Points

Mashcka [7]

Answer:

Step-by-step explanation:

Factor: -7x^2 -5x+18

Factor -1 out of 7x^2 -5x +18:

−(7x^2+5x−18)

Factor.

1. 18 is negative, so a negative number multiplied by a postive number.

2. You can't factor 7, so 7x times x.

(7x−9)(x+2)

4 0

4 years ago

This graph shows the tempature in Diego’s between moon and midnight one day. Choose ALL of the statements that are true.

tia_tia [17]

All of them are right

6 0

4 years ago