`P^^Q` means `(P+Q)/(Q-P)andPvvQ` means `(Q-P)/(Q+P)`
Answer the following questions:
`10^^2-7vv9`

Options

A. `-11/8`
B. `11/8`
C. `13/8`
D. `-13/8`

Answer

To solve the expression \(10 \uparrow 2 - 7 \downarrow 9\), we will first evaluate \(10 \uparrow 2\) and \(7 \downarrow 9\) separately using the definitions provided.

### Step 1: Calculate \(10 \uparrow 2\)

According to the definition, \(P \uparrow Q\) means \(\frac{P + Q}{Q - P}\).

Here, \(P = 10\) and \(Q = 2\).

Substituting the values into the formula:

\[ 10 \uparrow 2 = \frac{10 + 2}{2 - 10} \]

Calculating the numerator:

\[ 10 + 2 = 12 \]

Calculating the denominator:

\[ 2 - 10 = -8 \]

Now substituting back into the expression:

\[ 10 \uparrow 2 = \frac{12}{-8} = -\frac{3}{2} \]

### Step 2: Calculate \(7 \downarrow 9\)

According to the definition, \(P \downarrow Q\) means \(\frac{Q - P}{Q + P}\).

Here, \(P = 7\) and \(Q = 9\).

Substituting the values into the formula:

\[ 7 \downarrow 9 = \frac{9 - 7}{9 + 7} \]

Calculating the numerator:

\[ 9 - 7 = 2 \]

Calculating the denominator:

\[ 9 + 7 = 16 \]

Now substituting back into the expression:

\[ 7 \downarrow 9 = \frac{2}{16} = \frac{1}{8} \]

### Step 3: Combine the results

Now we need to subtract the two results we calculated:

\[ 10 \uparrow 2 - 7 \downarrow 9 = -\frac{3}{2} - \frac{1}{8} \]

To perform this subtraction, we need a common denominator. The least common multiple of 2 and 8 is 8.

Rewriting \(-\frac{3}{2}\) with a denominator of 8:

\[ -\frac{3}{2} = -\frac{3 \times 4}{2 \times 4} = -\frac{12}{8} \]

Now we can subtract:

\[ -\frac{12}{8} - \frac{1}{8} = -\frac{12 + 1}{8} = -\frac{13}{8} \]

### Final Answer

Thus, the final result for \(10 \uparrow 2 - 7 \downarrow 9\) is:

\[ -\frac{13}{8} \]

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