1. simplify the expression using the properties of real numbers. (-7 + 3)-|-2 + 9|-21-(-5) 2. simplify $\frac{3x}{y}-7x^{2}y^{-1}+2y^{2}x^{-1}$. 3. simplify $\frac{xy^{2}}{\frac{x^{0}x^{-3}}{(xy^{2})^{0}}\frac{xy^{-2}}{x^{4}}\frac{3y^{-2}}{x^{-4}}}$. 4. find the area of a circle with radius 1.5 inches. 5. find the percent change from 10 to 12.8. solve each equation. *6. 154 - 5(b)=16(4 - 6b)+10 *7. -2(2x - 3)-2^{2}-3=-x-(-4) for the following problems, x varies directly with y. find the constant of variation. *8. x = 12, y = 3 *9. x = 4, y = 16 10. simplify the expression 5x^{2}-2(x - 3x^{2}).

Question

Accepted Answer

### 1. Simplify the expression using the properties of real numbers: $(-7 + 3)-|-2 + 9|-21-(-5)$
# Explanation:
## Step1: Simplify inside parentheses
First, simplify $-7 + 3=-4$ and $-2 + 9 = 7$.
The expression becomes $-4-|7|-21 + 5$.
## Step2: Evaluate the absolute - value
Since $|7| = 7$, the expression is $-4-7-21 + 5$.
## Step3: Combine like - terms
$-4-7=-11$, and $-11-21=-32$, then $-32 + 5=-27$.

# Answer:
$-27$

### 2. Simplify $\frac{3x}{y}-7x^{2}y^{-1}+2y^{2}x^{-1}$
# Explanation:
## Step1: Rewrite with positive exponents
Recall that $y^{-1}=\frac{1}{y}$ and $x^{-1}=\frac{1}{x}$. The expression becomes $\frac{3x}{y}-\frac{7x^{2}}{y}+\frac{2y^{2}}{x}$.
## Step2: Find a common denominator
The common denominator of $y$ and $x$ is $xy$.
$\frac{3x\cdot x}{y\cdot x}-\frac{7x^{2}\cdot x}{y\cdot x}+\frac{2y^{2}\cdot y}{x\cdot y}=\frac{3x^{2}}{xy}-\frac{7x^{3}}{xy}+\frac{2y^{3}}{xy}$.
## Step3: Combine the fractions
$\frac{3x^{2}-7x^{3}+2y^{3}}{xy}$

# Answer:
$\frac{3x^{2}-7x^{3}+2y^{3}}{xy}$

### 3. Simplify $\frac{\frac{xy^{2}}{x^{0}x^{-3}}}{\frac{xy^{-2}}{(xy^{0})^{0}}-\frac{3y^{-2}}{x^{4}}}$
# Explanation:
## Step1: Use exponent rules
Recall that $x^{0} = 1$ and $(xy^{0})^{0}=1$. Also, $x^{-n}=\frac{1}{x^{n}}$.
$x^{0}x^{-3}=x^{-3}=\frac{1}{x^{3}}$, so $\frac{xy^{2}}{x^{0}x^{-3}}=x y^{2}\cdot x^{3}=x^{4}y^{2}$.
$xy^{-2}=\frac{x}{y^{2}}$ and $3y^{-2}=\frac{3}{y^{2}}$.
The denominator is $\frac{x}{y^{2}}-\frac{3}{x^{4}y^{2}}=\frac{x\cdot x^{4}-3}{x^{4}y^{2}}=\frac{x^{5}-3}{x^{4}y^{2}}$.
## Step2: Divide the fractions
$\frac{x^{4}y^{2}}{\frac{x^{5}-3}{x^{4}y^{2}}}=x^{4}y^{2}\cdot\frac{x^{4}y^{2}}{x^{5}-3}=\frac{x^{8}y^{4}}{x^{5}-3}$

# Answer:
$\frac{x^{8}y^{4}}{x^{5}-3}$

### 4. Find the area of a circle with radius $1.5$ inches
# Explanation:
## Step1: Recall the formula for the area of a circle
The formula for the area of a circle is $A=\pi r^{2}$, where $r$ is the radius.
## Step2: Substitute the value of $r$
Given $r = 1.5$ inches, then $A=\pi(1.5)^{2}=\pi\times2.25\approx3.14\times2.25 = 7.065$ square inches.

# Answer:
$7.065$ square inches

### 5. Find the percent change from $10$ to $12.8$
# Explanation:
## Step1: Recall the percent - change formula
The percent - change formula is $\text{Percent Change}=\frac{\text{New Value}-\text{Old Value}}{\text{Old Value}}\times100\%$.
## Step2: Substitute the values
New Value $=12.8$, Old Value $=10$.
$\frac{12.8 - 10}{10}\times100\%=\frac{2.8}{10}\times100\%=28\%$

# Answer:
$28\%$

### 6. Solve the equation $154-5b = 16(4 - 6b)+10$
# Explanation:
## Step1: Expand the right - hand side
$16(4 - 6b)+10=64-96b + 10=74-96b$.
The equation becomes $154-5b=74-96b$.
## Step2: Add $96b$ to both sides
$154-5b + 96b=74-96b+96b$, which simplifies to $154 + 91b=74$.
## Step3: Subtract $154$ from both sides
$91b=74 - 154=-80$.
## Step4: Solve for $b$
$b=-\frac{80}{91}$

# Answer:
$b =-\frac{80}{91}$

### 7. Solve the equation $-2(2x - 3)-2^{2}-3=-x-(-4)$
# Explanation:
## Step1: Simplify the left - hand side
$-2(2x - 3)-2^{2}-3=-4x + 6-4 - 3=-4x - 1$.
The right - hand side is $-x + 4$.
The equation becomes $-4x-1=-x + 4$.
## Step2: Add $4x$ to both sides
$-4x-1 + 4x=-x + 4+4x$, which simplifies to $-1 = 3x+4$.
## Step3: Subtract $4$ from both sides
$3x=-1 - 4=-5$.
## Step4: Solve for $x$
$x=-\frac{5}{3}$

# Answer:
$x =-\frac{5}{3}$

### 8. For the problem where $x$ varies directly with $y$, $x = 12,y = 3$. Find the constant of variation.
# Explanation:
## Step1: Recall the direct - variation formula
The formula for direct variation is $x=ky$, where $k$ is the constant of variation.
## Step2: Substitute the given values
Substitute $x = 12$ and $y = 3$ into $x=ky$, we get $12=k\times3$.
## Step3: Solve for $k$
$k = 4$

# Answer:
$k = 4$

### 9. For the problem where $x$ varies directly with $y$, $x = 4,y = 16$. Find the constant of variation.
# Explanation:
## Step1: Recall the direct - variation formula
The formula for direct variation is $x=ky$, where $k$ is the constant of variation.
## Step2: Substitute the given values
Substitute $x = 4$ and $y = 16$ into $x=ky$, we get $4=k\times16$.
## Step3: Solve for $k$
$k=\frac{4}{16}=\frac{1}{4}$

# Answer:
$k=\frac{1}{4}$

### 10. Simplify the expression $5x^{2}-2(x - 3x^{2})$
# Explanation:
## Step1: Distribute the $-2$
$-2(x - 3x^{2})=-2x+6x^{2}$.
The expression becomes $5x^{2}-2x + 6x^{2}$.
## Step2: Combine like - terms
$5x^{2}+6x^{2}-2x=11x^{2}-2x$

# Answer:
$11x^{2}-2x$

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QUESTION IMAGE

Question

Explanation:

1. Simplify the expression using the properties of real numbers: \((-7 + 3)-|-2 + 9|-21-(-5)\)

Step1: Simplify inside parentheses

Step2: Evaluate the absolute - value

Step3: Combine like - terms

Step1: Rewrite with positive exponents

Step2: Find a common denominator

Step3: Combine the fractions

Step1: Use exponent rules

Step2: Divide the fractions

Answer:

2. Simplify \(\frac{3x}{y}-7x^{2}y^{-1}+2y^{2}x^{-1}\)