13. solve: $r_mt_m = 160$, $r_pt_p = 400$, $r_p = 2r_m$, $t_p = t_m + 1$ simplify: 14. $x + \frac{x}{1 + \frac{1}{x}}$ 15. $a + \frac{b}{a + \frac{a}{b}}$ 16. $-3i^3 + 2\sqrt{-2}\sqrt{2} - \sqrt{-9}$ 17. $(-i - 1)(-3i + 2)$

Question

Accepted Answer

### Problem 13: Solve the system of equations
# Explanation:
## Step 1: Substitute $ R_P = 2R_M $ and $ T_P = T_M + 1 $ into $ R_PT_P = 400 $
We know $ R_P = 2R_M $ and $ T_P = T_M + 1 $, so substitute these into the equation $ R_PT_P = 400 $. We get $ (2R_M)(T_M + 1)=400 $. Also, we know from $ R_MT_M = 160 $ that $ R_M=\frac{160}{T_M} $ (assuming $ T_M
eq0 $).
## Step 2: Substitute $ R_M=\frac{160}{T_M} $ into the new equation
Substitute $ R_M=\frac{160}{T_M} $ into $ 2R_M(T_M + 1)=400 $. We have $ 2\times\frac{160}{T_M}\times(T_M + 1)=400 $. Simplify the left - hand side: $ \frac{320}{T_M}(T_M + 1)=400 $. Multiply both sides by $ T_M $ ( $ T_M
eq0 $) to get $ 320(T_M + 1)=400T_M $.
## Step 3: Expand and solve for $ T_M $
Expand the left - hand side: $ 320T_M+320 = 400T_M $. Subtract $ 320T_M $ from both sides: $ 320=400T_M - 320T_M $, which simplifies to $ 320 = 80T_M $. Then, divide both sides by 80: $ T_M=\frac{320}{80}=4 $.
## Step 4: Find $ R_M $
Since $ R_MT_M = 160 $ and $ T_M = 4 $, then $ R_M=\frac{160}{T_M}=\frac{160}{4}=40 $.
## Step 5: Find $ R_P $ and $ T_P $
Since $ R_P = 2R_M $, then $ R_P=2\times40 = 80 $. Since $ T_P=T_M + 1 $, then $ T_P=4 + 1=5 $.

# Answer: $ R_M = 40, T_M = 4, R_P = 80, T_P = 5 $

### Problem 14: Simplify $ x+\frac{x}{1+\frac{1}{x}} $
# Explanation:
## Step 1: Simplify the denominator of the fraction
First, simplify $ 1+\frac{1}{x} $. Find a common denominator, which is $ x $. So $ 1+\frac{1}{x}=\frac{x}{x}+\frac{1}{x}=\frac{x + 1}{x} $.
## Step 2: Simplify the complex fraction
Now, the expression $ \frac{x}{1+\frac{1}{x}} $ becomes $ \frac{x}{\frac{x + 1}{x}} $. When dividing by a fraction, we multiply by its reciprocal, so $ \frac{x}{\frac{x + 1}{x}}=x\times\frac{x}{x + 1}=\frac{x^{2}}{x + 1} $.
## Step 3: Add $ x $ to the simplified fraction
Now, we have $ x+\frac{x^{2}}{x + 1} $. Rewrite $ x $ with a common denominator of $ x + 1 $: $ x=\frac{x(x + 1)}{x + 1}=\frac{x^{2}+x}{x + 1} $. Then, $ x+\frac{x^{2}}{x + 1}=\frac{x^{2}+x}{x + 1}+\frac{x^{2}}{x + 1} $.
## Step 4: Combine the fractions
Combine the two fractions: $ \frac{x^{2}+x+x^{2}}{x + 1}=\frac{2x^{2}+x}{x + 1}=\frac{x(2x + 1)}{x + 1} $ (factoring the numerator).

# Answer: $ \frac{x(2x + 1)}{x + 1} $ (or $ \frac{2x^{2}+x}{x + 1} $)

### Problem 15: Simplify $ a+\frac{b}{a+\frac{a}{b}} $
# Explanation:
## Step 1: Simplify the denominator of the inner fraction
First, simplify $ a+\frac{a}{b} $. Find a common denominator of $ b $: $ a+\frac{a}{b}=\frac{ab}{b}+\frac{a}{b}=\frac{ab + a}{b}=\frac{a(b + 1)}{b} $.
## Step 2: Simplify the complex fraction
Now, the expression $ \frac{b}{a+\frac{a}{b}} $ becomes $ \frac{b}{\frac{a(b + 1)}{b}} $. Multiply by the reciprocal: $ \frac{b}{\frac{a(b + 1)}{b}}=b\times\frac{b}{a(b + 1)}=\frac{b^{2}}{a(b + 1)} $.
## Step 3: Add $ a $ to the simplified fraction
Rewrite $ a $ with a common denominator of $ a(b + 1) $: $ a=\frac{a\times a(b + 1)}{a(b + 1)}=\frac{a^{2}(b + 1)}{a(b + 1)} $. Then, $ a+\frac{b^{2}}{a(b + 1)}=\frac{a^{2}(b + 1)+b^{2}}{a(b + 1)} $.
## Step 4: Expand and simplify the numerator
Expand the numerator: $ a^{2}(b + 1)+b^{2}=a^{2}b+a^{2}+b^{2} $. So the simplified form is $ \frac{a^{2}b + a^{2}+b^{2}}{a(b + 1)} $.

# Answer: $ \frac{a^{2}b + a^{2}+b^{2}}{a(b + 1)} $ (or expand the numerator as needed)

### Problem 16: Simplify $ - 3i^{3}+2\sqrt{-2}\sqrt{2}-\sqrt{-9} $
# Explanation:
## Step 1: Recall the properties of $ i $ and square roots of negative numbers
Recall that $ i=\sqrt{-1} $, $ i^{2}=-1 $, $ i^{3}=i^{2}\times i=-i $, $ \sqrt{-2}=i\sqrt{2} $, and $ \sqrt{-9}=3i $.
## Step 2: Substitute these values into the expression
Substitute into the expression: $ -3i^{3}+2\sqrt{-2}\sqrt{2}-\sqrt{-9}=-3(-i)+2\times i\sqrt{2}\times\sqrt{2}-3i $.
## Step 3: Simplify each term
Simplify each term:
 - For $ -3(-i) $, we get $ 3i $.
 - For $ 2\times i\sqrt{2}\times\sqrt{2} $, $ \sqrt{2}\times\sqrt{2}=2 $, so $ 2\times i\times2 = 4i $.
 - For $ -3i $, it remains $ -3i $.
## Step 4: Combine like terms
Combine the terms: $ 3i + 4i-3i=(3 + 4-3)i = 4i $.

# Answer: $ 4i $

### Problem 17: Simplify $ (-i - 1)(-3i + 2) $
# Explanation:
## Step 1: Use the distributive property (FOIL method)
Use the FOIL method: $ (-i - 1)(-3i + 2)=(-i)\times(-3i)+(-i)\times2+(-1)\times(-3i)+(-1)\times2 $.
## Step 2: Simplify each term
Simplify each term:
 - $ (-i)\times(-3i)=3i^{2} $, and since $ i^{2}=-1 $, this is $ 3\times(-1)=-3 $.
 - $ (-i)\times2=-2i $.
 - $ (-1)\times(-3i)=3i $.
 - $ (-1)\times2=-2 $.
## Step 3: Combine like terms
Combine the terms: $ -3-2i + 3i-2=(-3-2)+(-2i + 3i)=-5 + i $.

# Answer: $ -5 + i $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 13: Solve the system of equations

Step 1: Substitute \( R_P = 2R_M \) and \( T_P = T_M + 1 \) into \( R_PT_P = 400 \)

Step 2: Substitute \( R_M=\frac{160}{T_M} \) into the new equation

Step 3: Expand and solve for \( T_M \)

Step 4: Find \( R_M \)

Step 5: Find \( R_P \) and \( T_P \)

Step 1: Simplify the denominator of the fraction

Step 2: Simplify the complex fraction

Step 3: Add \( x \) to the simplified fraction

Step 4: Combine the fractions

Step 1: Simplify the denominator of the inner fraction

Step 2: Simplify the complex fraction

Step 3: Add \( a \) to the simplified fraction

Step 4: Expand and simplify the numerator

Answer:

Problem 14: Simplify \( x+\frac{x}{1+\frac{1}{x}} \)