sptral review
subtract the polynomials.
1. $(7m^3 - 2m^2n + 6n) - (5m^2n - m^3 + 8n)$
2. $(2x^3y - 5y + 14x^2) - (6y + 7x^2 - 9x^3y)$
3. $(12x^2 - 5y + 16) - (7x^2 - 18y - 9)$
4. $(13xy + 7y - 16x^2) - (8y + 15xy - 7x^2)$
find the missing exponent.
1. $9.8 \times 10^6 + 7.5 \times 10^{\square} = 1.055 \times 10^7$
2. $35 \times 10^3 + 8.22 \times 10^{\square} = 4.322 \times 10^4$
3. $6.2 \text{ million} - 8.5 \times 10^{\square} = 61.15 \times 10^5$

Question

Accepted Answer

### Problem 1: Subtract the polynomials $(7m^3 - 2m^2n + 6n) - (5m^2n - m^3 + 8n)$
# Explanation:
## Step 1: Distribute the negative sign
We need to distribute the negative sign to each term inside the second parentheses:
$7m^3 - 2m^2n + 6n - 5m^2n + m^3 - 8n$

## Step 2: Combine like terms
- For $m^3$ terms: $7m^3 + m^3 = 8m^3$
- For $m^2n$ terms: $-2m^2n - 5m^2n = -7m^2n$
- For $n$ terms: $6n - 8n = -2n$

Putting it all together, we get: $8m^3 - 7m^2n - 2n$

# Answer:
$8m^3 - 7m^2n - 2n$

### Problem 2: Subtract the polynomials $(2x^3y - 5y + 14x^2) - (6y + 7x^2 - 9x^3y)$
# Explanation:
## Step 1: Distribute the negative sign
Distribute the negative sign to each term in the second polynomial:
$2x^3y - 5y + 14x^2 - 6y - 7x^2 + 9x^3y$

## Step 2: Combine like terms
- For $x^3y$ terms: $2x^3y + 9x^3y = 11x^3y$
- For $y$ terms: $-5y - 6y = -11y$
- For $x^2$ terms: $14x^2 - 7x^2 = 7x^2$

Putting it all together, we get: $11x^3y - 11y + 7x^2$

# Answer:
$11x^3y - 11y + 7x^2$

### Problem 3: Subtract the polynomials $(12x^2 - 5y + 16) - (7x^2 - 18y - 9)$
# Explanation:
## Step 1: Distribute the negative sign
Distribute the negative sign to each term in the second polynomial:
$12x^2 - 5y + 16 - 7x^2 + 18y + 9$

## Step 2: Combine like terms
- For $x^2$ terms: $12x^2 - 7x^2 = 5x^2$
- For $y$ terms: $-5y + 18y = 13y$
- For constant terms: $16 + 9 = 25$

Putting it all together, we get: $5x^2 + 13y + 25$

# Answer:
$5x^2 + 13y + 25$

### Problem 4: Subtract the polynomials $(13xy + 7y - 16x^2) - (8y + 15xy - 7x^2)$
# Explanation:
## Step 1: Distribute the negative sign
Distribute the negative sign to each term in the second polynomial:
$13xy + 7y - 16x^2 - 8y - 15xy + 7x^2$

## Step 2: Combine like terms
- For $xy$ terms: $13xy - 15xy = -2xy$
- For $y$ terms: $7y - 8y = -y$
- For $x^2$ terms: $-16x^2 + 7x^2 = -9x^2$

Putting it all together, we get: $-2xy - y - 9x^2$ (or $-9x^2 - 2xy - y$)

# Answer:
$-9x^2 - 2xy - y$

### Problem 1 (Find the missing exponent): $9.8 \times 10^6 + 7.5 \times 10^{\square} = 1.055 \times 10^7$
# Explanation:
## Step 1: Rewrite $1.055 \times 10^7$
We can rewrite $1.055 \times 10^7$ as $10.55 \times 10^6$ (since $1.055 \times 10 = 10.55$ and $10^7 = 10 \times 10^6$)

## Step 2: Subtract $9.8 \times 10^6$ from both sides
Let the missing exponent be $n$. So we have:
$7.5 \times 10^n = 10.55 \times 10^6 - 9.8 \times 10^6$
$7.5 \times 10^n = (10.55 - 9.8) \times 10^6$
$7.5 \times 10^n = 0.75 \times 10^6$

## Step 3: Simplify $0.75 \times 10^6$
We can rewrite $0.75 \times 10^6$ as $7.5 \times 10^5$ (since $0.75 \times 10 = 7.5$ and $10^6 = 10 \times 10^5$)

So, $7.5 \times 10^n = 7.5 \times 10^5$, which means $n = 5$

# Answer:
$5$

### Problem 2 (Find the missing exponent): $35 \times 10^3 + 8.22 \times 10^{\square} = 4.322 \times 10^4$
# Explanation:
## Step 1: Rewrite $4.322 \times 10^4$
We can rewrite $4.322 \times 10^4$ as $43.22 \times 10^3$ (since $4.322 \times 10 = 43.22$ and $10^4 = 10 \times 10^3$)

## Step 2: Subtract $35 \times 10^3$ from both sides
Let the missing exponent be $n$. So we have:
$8.22 \times 10^n = 43.22 \times 10^3 - 35 \times 10^3$
$8.22 \times 10^n = (43.22 - 35) \times 10^3$
$8.22 \times 10^n = 8.22 \times 10^3$

So, $n = 3$

# Answer:
$3$

### Problem 3 (Find the missing exponent): $6.2$ million $ - 8.5 \times 10^{\square} = 61.15 \times 10^5$
# Explanation:
## Step 1: Rewrite $6.2$ million and $61.15 \times 10^5$
- $6.2$ million is $6.2 \times 10^6$ or $62 \times 10^5$
- $61.15 \times 10^5$ remains as is.

## Step 2: Subtract $61.15 \times 10^5$ from $62 \times 10^5$
Let the missing exponent be $n$. So we have:
$8.5 \times 10^n = 62 \times 10^5 - 61.15 \times 10^5$
$8.5 \times 10^n = (62 - 61.15) \times 10^5$
$8.5 \times 10^n = 0.85 \times 10^5$

## Step 3: Simplify $0.85 \times 10^5$
We can rewrite $0.85 \times 10^5$ as $8.5 \times 10^4$ (since $0.85 \times 10 = 8.5$ and $10^5 = 10 \times 10^4$)

So, $8.5 \times 10^n = 8.5 \times 10^4$, which means $n = 4$

# Answer:
$4$

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QUESTION IMAGE

Question

Explanation:

Problem 1: Subtract the polynomials \((7m^3 - 2m^2n + 6n) - (5m^2n - m^3 + 8n)\)

Step 1: Distribute the negative sign

Step 2: Combine like terms

Step 1: Distribute the negative sign

Step 2: Combine like terms

Step 1: Distribute the negative sign

Step 2: Combine like terms

Answer:

Problem 2: Subtract the polynomials \((2x^3y - 5y + 14x^2) - (6y + 7x^2 - 9x^3y)\)