Question

1. the volume of the rectangular prism below is $36x^6y^8$. determine an expression to represent the missing side.
2. which expression is equivalent to $\frac{16k^9m^7}{4k^3m^{14}}$ for all values of $k$ and $m$ where the expression is defined?

a. $\frac{12k^6}{m^7}$
b. $\frac{4k^6}{m^7}$
c. $\frac{12k^3}{m^2}$
d. $\frac{4k^3}{m^2}$

1. the area of a rectangle is $54x^9y^8$ square yards. if the length of the rectangle is $6x^3y^4$ yards, which expression represents the width of the rectangle in yards?

a. $9x^3y^2$
b. $48x^6y^4$
c. $9x^6y^4$
d. $60x^{12}y^{12}$

Explanation:

Question 16 (Volume of Rectangular Prism)

Step 1: Recall Volume Formula

The volume \( V \) of a rectangular prism is given by \( V = l \times w \times h \), where \( l \), \( w \), and \( h \) are the length, width, and height. Let the missing side be \( s \). We know \( V = 36x^6y^8 \), one side is \( 3xy \), and another is \( 4x^2y^5 \). So, \( 36x^6y^8 = 3xy \times 4x^2y^5 \times s \).

Step 2: Simplify the Known Product

First, multiply \( 3xy \) and \( 4x^2y^5 \):
\( 3xy \times 4x^2y^5 = (3 \times 4) \times (x \times x^2) \times (y \times y^5) = 12x^{1 + 2}y^{1 + 5} = 12x^3y^6 \).

Step 3: Solve for \( s \)

Now, divide the volume by \( 12x^3y^6 \) to find \( s \):
\( s = \frac{36x^6y^8}{12x^3y^6} \).
Simplify the coefficients and exponents:

• Coefficient: \( \frac{36}{12} = 3 \)
• \( x \)-exponent: \( x^{6 - 3} = x^3 \)
• \( y \)-exponent: \( y^{8 - 6} = y^2 \)

So, \( s = 3x^3y^2 \)? Wait, no—wait, maybe I misread the given sides. Wait, the diagram shows \( 3xy \) and \( 4x^2y^5 \), volume \( 36x^6y^8 \). Wait, let's recalculate:

Wait, \( 3xy \times 4x^2y^5 = 12x^3y^6 \). Then \( 36x^6y^8 \div 12x^3y^6 = 3x^{6 - 3}y^{8 - 6} = 3x^3y^2 \). But maybe the problem has different given sides? Wait, maybe the first side is \( 3y \) (not \( 3xy \))? Wait, the diagram says \( 3y \) (maybe a typo, or I misread). Let me check again. If the sides are \( 3y \), \( 4x^2y^5 \), and volume \( 36x^6y^8 \), then:

\( 3y \times 4x^2y^5 = 12x^2y^6 \). Then \( 36x^6y^8 \div 12x^2y^6 = 3x^{6 - 2}y^{8 - 6} = 3x^4y^2 \)? No, this is confusing. Wait, maybe the correct approach is:

Volume of rectangular prism: \( V = l \times w \times h \). Let the missing side be \( s \). Given \( V = 36x^6y^8 \), \( l = 3y \), \( w = 4x^2y^5 \). So:

\( 36x^6y^8 = 3y \times 4x^2y^5 \times s \)
\( 36x^6y^8 = 12x^2y^6 \times s \)
\( s = \frac{36x^6y^8}{12x^2y^6} = 3x^{4}y^{2} \)? Wait, no, maybe the first side is \( 3x \) instead of \( 3y \). Let's assume the sides are \( 3x \), \( 4x^2y^5 \), and volume \( 36x^6y^8 \):

\( 3x \times 4x^2y^5 = 12x^3y^5 \). Then \( 36x^6y^8 \div 12x^3y^5 = 3x^3y^3 \). Hmm, maybe I made a mistake. Alternatively, maybe the correct answer is derived as:

Wait, let's do it properly. Let the three sides be \( a = 3xy \), \( b = 4x^2y^5 \), \( c = s \). Then \( a \times b \times c = 36x^6y^8 \).

\( a \times b = 3xy \times 4x^2y^5 = 12x^3y^6 \). Then \( c = 36x^6y^8 / 12x^3y^6 = 3x^3y^2 \). Wait, but maybe the problem has \( 3y \) and \( 4x^2y^5 \), so \( 3y \times 4x^2y^5 = 12x^2y^6 \), then \( 36x^6y^8 / 12x^2y^6 = 3x^4y^2 \). I think the intended answer is \( 3x^3y^2 \) or similar, but maybe I misread the problem. Let's proceed with the given numbers.

Question 17 (Simplify Rational Expression)

Step 1: Simplify Coefficients and Exponents Separately

We have \( \frac{16k^9m^7}{4k^3m^{14}} \).

• Coefficients: \( \frac{16}{4} = 4 \).
• \( k \)-exponents: \( k^{9 - 3} = k^6 \).
• \( m \)-exponents: \( m^{7 - 14} = m^{-7} = \frac{1}{m^7} \) (but wait, no—wait, the expression is \( \frac{16k^9m^7}{4k^3m^{14}} \), so exponents subtract: \( 9 - 3 = 6 \) for \( k \), \( 7 - 14 = -7 \) for \( m \), so \( m^{-7} = \frac{1}{m^7} \). Thus, the expression becomes \( 4 \times k^6 \times \frac{1}{m^7} = \frac{4k^6}{m^7} \), which is option B.

Question 18 (Area of Rectangle)

Step 1: Recall Area Formula

The area \( A \) of a rectangle is \( A = l \times w \), where \( l \) is length and \( w \) is width. We know \( A = 54x^9y^8 \) (wait, the problem says "54x⁹y⁸ square yards" and length is \( 6x^3y^4 \). Wait, the problem states: "The area of a rectangle is 54x⁹y⁸ square yards. If the length of the rectangle is 6x³y⁴ yards, which expression represents the width of the rectangle in yards?"

Step 2: Solve for Width

Width \( w = \frac{A}{l} = \frac{54x^9y^8}{6x^3y^4} \).

Step 3: Simplify

• Coefficient: \( \frac{54}{6} = 9 \).
• \( x \)-exponent: \( x^{9 - 3} = x^6 \).
• \( y \)-exponent: \( y^{8 - 4} = y^4 \).

Thus, \( w = 9x^6y^4 \), which is option C.

Final Answers:

1. (Assuming correct calculation) Let's recheck: If \( V = 36x^6y^8 \), sides \( 3xy \) and \( 4x^2y^5 \), then:

\( 3xy \times 4x^2y^5 = 12x^3y^6 \), so \( s = 36x^6y^8 / 12x^3y^6 = 3x^3y^2 \). But maybe the problem has a typo, but following the process, the missing side is \( 3x^3y^2 \) (not in options? Wait, maybe I misread the volume. Wait, the volume is 36x⁶y⁸, sides 3y and 4x²y⁵:
\( 3y \times 4x²y⁵ = 12x²y⁶ \), so \( s = 36x⁶y⁸ / 12x²y⁶ = 3x⁴y² \). Hmm, maybe the options are different. Alternatively, maybe the first side is 3x, second 4x²y⁵, volume 36x⁶y⁸:
\( 3x \times 4x²y⁵ = 12x³y⁵ \), \( s = 36x⁶y⁸ / 12x³y⁵ = 3x³y³ \). Not matching. Maybe the problem has a different setup, but proceeding with the given options, perhaps the intended answer is \( 3x^3y^2 \) (not listed, so maybe I made a mistake).

1. \( \frac{16k^9m^7}{4k^3m^{14}} = \frac{16}{4} \times k^{9 - 3} \times m^{7 - 14} = 4k^6m^{-7} = \frac{4k^6}{m^7} \), so B. \( \frac{4k^6}{m^7} \).

1. \( \frac{54x^9y^8}{6x^3y^4} = 9x^6y^4 \), so C. \( 9x^6y^4 \).

Final Answers (Boxed for Each):

1. (Assuming correct process, but options may vary) Let's confirm 17 and 18:

1. \(\boxed{\frac{4k^6}{m^7}}\) (Option B)

1. \(\boxed{9x^6y^4}\) (Option C)

Answer:

Step 1: Recall Area Formula

The area \( A \) of a rectangle is \( A = l \times w \), where \( l \) is length and \( w \) is width. We know \( A = 54x^9y^8 \) (wait, the problem says "54x⁹y⁸ square yards" and length is \( 6x^3y^4 \). Wait, the problem states: "The area of a rectangle is 54x⁹y⁸ square yards. If the length of the rectangle is 6x³y⁴ yards, which expression represents the width of the rectangle in yards?"

Step 2: Solve for Width

Width \( w = \frac{A}{l} = \frac{54x^9y^8}{6x^3y^4} \).

Step 3: Simplify

• Coefficient: \( \frac{54}{6} = 9 \).
• \( x \)-exponent: \( x^{9 - 3} = x^6 \).
• \( y \)-exponent: \( y^{8 - 4} = y^4 \).

Thus, \( w = 9x^6y^4 \), which is option C.

Final Answers:

1. (Assuming correct calculation) Let's recheck: If \( V = 36x^6y^8 \), sides \( 3xy \) and \( 4x^2y^5 \), then:

\( 3xy \times 4x^2y^5 = 12x^3y^6 \), so \( s = 36x^6y^8 / 12x^3y^6 = 3x^3y^2 \). But maybe the problem has a typo, but following the process, the missing side is \( 3x^3y^2 \) (not in options? Wait, maybe I misread the volume. Wait, the volume is 36x⁶y⁸, sides 3y and 4x²y⁵:
\( 3y \times 4x²y⁵ = 12x²y⁶ \), so \( s = 36x⁶y⁸ / 12x²y⁶ = 3x⁴y² \). Hmm, maybe the options are different. Alternatively, maybe the first side is 3x, second 4x²y⁵, volume 36x⁶y⁸:
\( 3x \times 4x²y⁵ = 12x³y⁵ \), \( s = 36x⁶y⁸ / 12x³y⁵ = 3x³y³ \). Not matching. Maybe the problem has a different setup, but proceeding with the given options, perhaps the intended answer is \( 3x^3y^2 \) (not listed, so maybe I made a mistake).

1. \( \frac{16k^9m^7}{4k^3m^{14}} = \frac{16}{4} \times k^{9 - 3} \times m^{7 - 14} = 4k^6m^{-7} = \frac{4k^6}{m^7} \), so B. \( \frac{4k^6}{m^7} \).

1. \( \frac{54x^9y^8}{6x^3y^4} = 9x^6y^4 \), so C. \( 9x^6y^4 \).

Final Answers (Boxed for Each):

1. (Assuming correct process, but options may vary) Let's confirm 17 and 18:

1. \(\boxed{\frac{4k^6}{m^7}}\) (Option B)

1. \(\boxed{9x^6y^4}\) (Option C)