Question

Can someone please show me the steps in finding the answers to these two problems ?

-thank you so much !

Answer 1

`\displaystyle 24 = x \\ 8^(x - 1) = y`

Step-by-step explanation:

`\displaystyle [(\sqrt[3]{9})(\sqrt[3]{x})]^3 = 6^3 → 216 = 9x; 24 = x`

According to the Definition of Rational Exponents [part II], you can rewrite an exponents as radicals:

`\displaystyle \sqrt[n]{a}^m = a^{(m)/(n)} \\ \\ \sqrt[3]{x} = x^{(1)/(3)} \\ \sqrt[3]{9} = 9^{(1)/(3)}`

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This table is in the form of a geometric sequence:

`\displaystyle a_n = a_1r^(n - 1), where\:n\:is\:the\:common\:ratio \\ \\ y = 8^(x - 1)`

Since 8 is the common ratio, all you have to do is plug in each value from the table, into the function to confirm their authenticities.

I am joyous to assist you anytime.

Answer 2

• y = 8^(x-1)
• x = 24

Explanation:

The first step is always to look at the question to see what you are given and what you are asked for. The next step is to identify the relevant relations between what you have and what you need to find.

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1) You are given a sequence of (x, y) points where x increases by 1 from one point to the next, and y increases by a factor of 8. This tells you the value of y form a geometric sequence that is described by an exponential function.

The general form of the n-th term of a geometric sequence is ...

an = a1·r^(n-1)

where a1 is the value for n=1, and r is the common ratio.

If this were a geometric sequence, we could write the general term using a1=1 and r=8:

an = 1·8^(n-1)

Instead, this is a relation between x and y. It is the same relation as for the geometric sequence, but with different variable names. We have y instead of an, and we have x instead of n. So, the equation is ...

y = 8^(x -1) . . . . . . . or . . y = (1/8)8^x

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2) This equation is of the form ...

k·∛x = 6

Since k is also a cube root, it is easiest to cube both sides of the equation:

9x = 6³ = 216

Now, you can divide by 9 to find x:

x = 216/9

x = 24