integrated 2
d2 review preview
name:
date:
per:
1. rewrite each of the expressions below. your final expressions should not contain negative exponents or parentheses. homework help

a. $(5x^3)(- 3x^{-2})$

b. $(4p^2q)^3$

c. $\frac{3m^7}{m^{-1}}$

2. use the information given for each diagram below to solve for x. show all work. homework help

a. $\vec{bd}$ bisects $\angle abc$. (remember that an angle bisector divides the angle into two congruent angles.) if $m\angle abd = 5x - 10^\circ$ and $m\angle abc = 65^\circ$, solve for x.

b. point m is a midpoint of $\overline{ef}$. if $\overline{em} = 4x - 2$ and $\overline{mf} = 3x + 9$, solve for x.

c. solve for x and y.

Question

integrated 2
d2 review preview
name:
date:
per:
1. rewrite each of the expressions below. your final expressions should not contain negative exponents or parentheses. homework help

a. $(5x^3)(- 3x^{-2})$

b. $(4p^2q)^3$

c. $\frac{3m^7}{m^{-1}}$

2. use the information given for each diagram below to solve for x. show all work. homework help

a. $\vec{bd}$ bisects $\angle abc$. (remember that an angle bisector divides the angle into two congruent angles.) if $m\angle abd = 5x - 10^\circ$ and $m\angle abc = 65^\circ$, solve for x.

b. point m is a midpoint of $\overline{ef}$. if $\overline{em} = 4x - 2$ and $\overline{mf} = 3x + 9$, solve for x.

c. solve for x and y.

Accepted Answer

### Problem 1a:
# Explanation:
## Step1: Multiply coefficients and variables
Multiply the coefficients $5$ and $-3$, and use the exponent rule $a^m \cdot a^n = a^{m + n}$ for $x$:
$(5x^3)(-3x^{-2}) = 5 \cdot (-3) \cdot x^{3 + (-2)}$

## Step2: Simplify
Calculate $5 \cdot (-3) = -15$ and $x^{3 + (-2)} = x^1 = x$:
$-15x$

### Problem 1b:
# Explanation:
## Step1: Apply power of a product rule
Use $(ab)^n = a^n b^n$ and $(a^m)^n = a^{m \cdot n}$:
$(4p^2 q)^3 = 4^3 \cdot (p^2)^3 \cdot q^3$

## Step2: Simplify exponents and coefficients
Calculate $4^3 = 64$, $(p^2)^3 = p^{2 \cdot 3} = p^6$:
$64p^6 q^3$

### Problem 1c:
# Explanation:
## Step1: Use quotient rule for exponents
Use $ \frac{a^m}{a^n} = a^{m - n} $ (and $a^{-n} = \frac{1}{a^n}$, so $ \frac{1}{m^{-1}} = m^1 $):
$\frac{3m^7}{m^{-1}} = 3 \cdot m^{7 - (-1)}$

## Step2: Simplify the exponent
Calculate $7 - (-1) = 8$:
$3m^8$

### Problem 2a:
# Explanation:
## Step1: Use angle bisector definition
Since $\overrightarrow{BD}$ bisects $\angle ABC$, $m\angle ABC = 2 \cdot m\angle ABD$. So:
$65^\circ = 2(5x - 10^\circ)$

## Step2: Solve for $x$
Divide both sides by $2$: $32.5^\circ = 5x - 10^\circ$
Add $10^\circ$ to both sides: $42.5^\circ = 5x$
Divide by $5$: $x = 8.5$

### Problem 2b:
# Explanation:
## Step1: Use midpoint definition
Since $M$ is the midpoint of $\overline{EF}$, $EM = MF$. So:
$4x - 2 = 3x + 9$

## Step2: Solve for $x$
Subtract $3x$ from both sides: $x - 2 = 9$
Add $2$ to both sides: $x = 11$

### Problem 2c:
# Explanation:
## Step1: Find $x$ (supplementary angles)
The angle adjacent to $127^\circ$ is $180^\circ - 127^\circ = 53^\circ$. Since $x$ and $83^\circ$ are related by parallel lines (corresponding angles or vertical angles), wait—actually, the angle with $83^\circ$ and $x$: Wait, the first transversal: the angle $83^\circ$ and $x$? Wait, no—wait, the two transversals are parallel? Wait, the angle $127^\circ$ and its supplementary angle is $53^\circ$, and $x$ should be equal to $83^\circ$? Wait, no, let's re-examine. Wait, the first angle is $83^\circ$, and the angle $x$ and the angle supplementary to $127^\circ$ (which is $53^\circ$)? Wait, no, maybe vertical angles or corresponding angles. Wait, actually, the angle with $83^\circ$ and $x$: Wait, the two lines (the transversals) are parallel? Wait, the angle $127^\circ$ and the angle adjacent to $y$: Wait, let's do $x$ first. The angle $127^\circ$ and its linear pair is $180 - 127 = 53^\circ$. Then, the angle $x$ and $83^\circ$: Wait, no, maybe $x = 83^\circ$ (corresponding angles) and $y = 53^\circ$? Wait, no, let's solve properly.

Wait, for $x$: The angle $83^\circ$ and $x$ are corresponding angles (if the lines are parallel), so $x = 83^\circ$? Wait, no, the other angle: the angle supplementary to $127^\circ$ is $53^\circ$, and $y$ should be equal to $53^\circ$? Wait, maybe I made a mistake. Let's correct:

First, the angle adjacent to $127^\circ$ is $180^\circ - 127^\circ = 53^\circ$. Now, the angle $y$ and this $53^\circ$ angle are corresponding angles (if the two transversals are parallel), so $y = 53^\circ$. For $x$, the angle $83^\circ$ and $x$ are corresponding angles, so $x = 83^\circ$? Wait, no, maybe $x$ is equal to $180^\circ - 83^\circ = 97^\circ$? No, let's use the diagram. Wait, the first transversal has an $83^\circ$ angle, and the second transversal has a $127^\circ$ angle. The angle $x$ and the angle supplementary to $127^\circ$ (53°) and $83^\circ$: Wait, maybe $x + 83^\circ = 180^\circ$? No, that would be $x = 97^\circ$, but that doesn't match. Wait, perhaps the correct approach is:

For $x$: The angle $83^\circ$ and $x$ are same-side interior angles? No, maybe vertical angles. Wait, I think I messed up. Let's start over.

The angle $127^\circ$ and its linear pair is $53^\circ$ (since $180 - 127 = 53$). Now, the angle $y$ and this $53^\circ$ angle are vertical angles or corresponding angles, so $y = 53^\circ$. For $x$, the angle $83^\circ$ and $x$ are same-side interior angles? No, maybe $x = 83^\circ$ (corresponding angles). Wait, maybe the correct answers are $x = 83^\circ$ and $y = 53^\circ$. But let's check:

If the two transversals (the slanted lines) are parallel, then the angle $83^\circ$ and $x$ are corresponding angles, so $x = 83^\circ$. The angle $y$ and the angle supplementary to $127^\circ$ (53°) are corresponding angles, so $y = 53^\circ$.

# Answers:
1a. $\boldsymbol{-15x}$
1b. $\boldsymbol{64p^6 q^3}$
1c. $\boldsymbol{3m^8}$
2a. $\boldsymbol{x = 8.5}$
2b. $\boldsymbol{x = 11}$
2c. $\boldsymbol{x = 83^\circ}$, $\boldsymbol{y = 53^\circ}$ (assuming corresponding angles)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1a:

Step1: Multiply coefficients and variables

Step2: Simplify

Problem 1b:

Step1: Apply power of a product rule

Step2: Simplify exponents and coefficients

Problem 1c:

Step1: Use quotient rule for exponents

Step2: Simplify the exponent

Problem 2a:

Answer: