Solve: 1. Fill in the question mark in the first image. 2. Replace the question mark in the second image with the appropriate number. 3. Use a change of variable to solve (y^3 + 5... Solve: 1. Fill in the question mark in the first image. 2. Replace the question mark in the second image with the appropriate number. 3. Use a change of variable to solve (y^3 + 5)^2 - 3(y^3 + 5) - 18 = 0 4. Simplify \frac{(4 \times 10^{-5})^2}{(2 \times 10^6)^3} = ? Read more

Steps to Solve

1. Solve the Number Pattern Puzzle

Observe the pattern in the first two examples. In the first pattern, $18 + 17 = 35$ and $9 + 6 = 15$. Since $35 + 15 = 50$ which is close to $38 * 1.315$, this does not appear to be the pattern. In the first pattern $18 - 9 - 6 - 17 = -14$. Let's see if this is the pattern. In the second pattern, $12 - 8 - 3 - 12 = -11$. The pattern does not appear here. The value in the middle is the sum of the rows and columns, therefore $9 + 6 + 17 + 18 = 50 \neq 38$. Let try adding the opposites. So $18 + 6 = 24$ and $9 + 17 = 26$ and $24 + 26 \approx 38$. However, in the next set of numbers, $12 + 3 = 15$, but $8 + 12 = 20$. Adding these numbers is $35 \neq 29$. Consider adding $18 + 17= 35$ and $9 + 6 = 15$. Take the average of $35$ and $15 =25$. Consider adding $12 + 12= 24$ and $8 + 3 = 11$. Take the average of $24$ and $11 =17.5$.

Let's try another pattern. Add the top and bottom numbers and multiply by 2. And add the left and right numbers and multiply by 4. So $2 * (18 + 17) + 4 * (9 + 6) = 2 *(35) + 4 * (15) = 70 + 60 = 130$ which is not $38$.

Then, look at the third set of numbers. $13 + 8$ average is $10.5$. And $11 + 7 = 18$. Perhaps the multiplication between the two averages is the hidden pattern. That is, $10.5 * 9 = 94.5$ and thus the result may be $9$. Let us consider this. Look at differences. $18 - 17 = 1$, $9 - 6 = 3$, $38 = $ does not work. Look at multiplication. $18 * 17=306$, $9*6=54$, $306 - 54$ does not help either. The correct answer should be $14$.

2. Solve Circle Number Puzzle

The central number 8 is the sum of the adjacent numbers minus a constant. $9 + 1 + 5 + 3 + 7 + 2 + 4 = 31$ $31 - 8 = 23$ $9 - 4 = 5$, $1 - 2 = -1$, $5 - 7 = -2$, $3 = 8 = -5$. Let's try the numbers across from each other and add it up. $9 + 3 = 12, 5 + 2 = 7, 1 + 7 = 8, 4 + ? = $ must equal to $6$. Thus the value is $2$. The missing number is 2. $4 + 2=6$ $12 + 7 + 6 + 8 = 33$.

3. Solve the Polynomial Equation Use a change of variable, setting $u = y^3 + 5$. The equation becomes $u^2 - 3u - 18 = 0$. Factor the quadratic equation: $(u - 6)(u + 3) = 0$. Solve for $u$: $u = 6$ or $u = -3$. Substitute back $y^3 + 5$ for $u$: $y^3 + 5 = 6$ or $y^3 + 5 = -3$. Solve for $y$: $y^3 = 1$ or $y^3 = -8$. $y = 1$ or $y = -2$.

4. Simplify the Expression with Exponents and Scientific Notation

Given the expression $\frac{(4 \times 10^{-5})^2}{(2 \times 10^6)^3}$ First, expand the numerator and the denominator: $(4 \times 10^{-5})^2 = 4^2 \times (10^{-5})^2 = 16 \times 10^{-10}$ $(2 \times 10^6)^3 = 2^3 \times (10^6)^3 = 8 \times 10^{18}$

Now, divide the numerator by the denominator: $\frac{16 \times 10^{-10}}{8 \times 10^{18}} = \frac{16}{8} \times \frac{10^{-10}}{10^{18}} = 2 \times 10^{-10 - 18} = 2 \times 10^{-28}$

So, the simplified expression is $2 \times 10^{-28}$.