simplify each expression.
1) $dfrac{dfrac{5}{4} + dfrac{m + 1}{4}}{5}$
2) $dfrac{dfrac{u}{4} + dfrac{4}{25}}{25}$
3) $dfrac{dfrac{4}{5} - dfrac{25}{x^2}}{16}$
4) $dfrac{dfrac{m + 5}{2}}{dfrac{m}{4} + dfrac{1}{m + 5}}$

Question

Accepted Answer

### Problem 1:
# Explanation:
## Step1: Simplify the numerator
First, simplify the numerator of the complex fraction. The numerator is $\frac{5}{4}+\frac{m + 1}{4}$. Since the denominators are the same, we can add the numerators:
$\frac{5+(m + 1)}{4}=\frac{5+m + 1}{4}=\frac{m+6}{4}$

## Step2: Divide by the denominator
Now, we have the complex fraction $\frac{\frac{m + 6}{4}}{5}$. Dividing by a number is the same as multiplying by its reciprocal, so we multiply $\frac{m + 6}{4}$ by $\frac{1}{5}$:
$\frac{m + 6}{4}	imes\frac{1}{5}=\frac{m + 6}{20}$

# Answer:
$\frac{m + 6}{20}$

### Problem 2:
# Explanation:
## Step1: Simplify the numerator
The numerator of the complex fraction is $\frac{u}{4}+\frac{4}{25}$. To add these fractions, we find a common denominator, which is $4	imes25 = 100$. Then we rewrite the fractions with the common denominator:
$\frac{u	imes25}{4	imes25}+\frac{4	imes4}{25	imes4}=\frac{25u}{100}+\frac{16}{100}=\frac{25u + 16}{100}$

## Step2: Divide by the denominator
Now we have the complex fraction $\frac{\frac{25u+16}{100}}{25}$. Dividing by 25 is the same as multiplying by $\frac{1}{25}$:
$\frac{25u + 16}{100}	imes\frac{1}{25}=\frac{25u+16}{2500}$

# Answer:
$\frac{25u + 16}{2500}$

### Problem 3:
# Explanation:
## Step1: Simplify the numerator
The numerator of the complex fraction is $\frac{4}{5}-\frac{25}{x^{2}}$. Find a common denominator, which is $5x^{2}$. Rewrite the fractions:
$\frac{4	imes x^{2}}{5	imes x^{2}}-\frac{25	imes5}{x^{2}	imes5}=\frac{4x^{2}}{5x^{2}}-\frac{125}{5x^{2}}=\frac{4x^{2}-125}{5x^{2}}$

## Step2: Divide by the denominator
Now we have the complex fraction $\frac{\frac{4x^{2}-125}{5x^{2}}}{16}$. Dividing by 16 is the same as multiplying by $\frac{1}{16}$:
$\frac{4x^{2}-125}{5x^{2}}	imes\frac{1}{16}=\frac{4x^{2}-125}{80x^{2}}$

# Answer:
$\frac{4x^{2}-125}{80x^{2}}$

### Problem 4:
# Explanation:
## Step1: Simplify the denominator
First, simplify the denominator of the complex fraction, which is $\frac{m}{4}+\frac{1}{m + 5}$. Find a common denominator, which is $4(m + 5)$. Rewrite the fractions:
$\frac{m(m + 5)}{4(m + 5)}+\frac{1	imes4}{(m + 5)	imes4}=\frac{m(m + 5)+4}{4(m + 5)}$
Expand the numerator: $m(m + 5)+4=m^{2}+5m + 4$
Factor the numerator: $m^{2}+5m + 4=(m + 1)(m + 4)$ (Wait, actually $m^{2}+5m + 4=(m + 1)(m + 4)$? Wait, no, $m^{2}+5m + 4=(m + 1)(m + 4)$? Wait, $1	imes4 = 4$ and $1 + 4=5$, yes. So the denominator becomes $\frac{(m + 1)(m + 4)}{4(m + 5)}$

## Step2: Divide the numerator by the denominator
The numerator of the complex fraction is $\frac{m + 5}{2}$. Dividing by the denominator $\frac{(m + 1)(m + 4)}{4(m + 5)}$ is the same as multiplying by its reciprocal:
$\frac{m + 5}{2}	imes\frac{4(m + 5)}{(m + 1)(m + 4)}$
Simplify the constants: $\frac{4}{2}=2$, so we have $2	imes\frac{(m + 5)(m + 5)}{(m + 1)(m + 4)}=\frac{2(m + 5)^{2}}{(m + 1)(m + 4)}$ (Wait, wait, let's check again. Wait, the denominator after simplifying is $\frac{m(m + 5)+4}{4(m + 5)}=\frac{m^{2}+5m + 4}{4(m + 5)}=\frac{(m + 1)(m + 4)}{4(m + 5)}$. Then the numerator is $\frac{m + 5}{2}$. So multiplying by reciprocal: $\frac{m + 5}{2}	imes\frac{4(m + 5)}{(m + 1)(m + 4)}$. The 4 and 2 simplify to 2, so we get $\frac{2(m + 5)(m + 5)}{(m + 1)(m + 4)}=\frac{2(m + 5)^{2}}{(m^{2}+5m + 4)}$

Wait, maybe I made a mistake in factoring. Wait, $m^{2}+5m + 4$: factors of 4 that add to 5 are 1 and 4, so yes, $(m + 1)(m + 4)$. So the final simplification is $\frac{2(m + 5)^{2}}{(m + 1)(m + 4)}$ or we can expand the numerator: $2(m^{2}+10m + 25)=2m^{2}+20m + 50$, and the denominator is $m^{2}+5m + 4$. But the factored form is probably better.

Wait, let's redo step 1 for the denominator:

Denominator: $\frac{m}{4}+\frac{1}{m + 5}=\frac{m(m + 5)+4}{4(m + 5)}=\frac{m^{2}+5m + 4}{4(m + 5)}=\frac{(m + 1)(m + 4)}{4(m + 5)}$. Correct.

Numerator: $\frac{m + 5}{2}$

So dividing numerator by denominator: $\frac{m + 5}{2}\div\frac{(m + 1)(m + 4)}{4(m + 5)}=\frac{m + 5}{2}	imes\frac{4(m + 5)}{(m + 1)(m + 4)}$

Simplify 4/2 = 2, so:

$2	imes\frac{(m + 5)(m + 5)}{(m + 1)(m + 4)}=\frac{2(m + 5)^{2}}{(m + 1)(m + 4)}$

Yes, that's correct.

# Answer:
$\frac{2(m + 5)^{2}}{(m + 1)(m + 4)}$ (or expanded as $\frac{2m^{2}+20m + 50}{m^{2}+5m + 4}$)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1:

Step1: Simplify the numerator

Step2: Divide by the denominator

Step1: Simplify the numerator

Step2: Divide by the denominator

Step1: Simplify the numerator

Step2: Divide by the denominator

Answer:

Problem 2: