Question

1. \\(\frac{2^{-1}}{5^0}\\) 6. \\(\frac{-3^{-2}}{2^{-1}}\\) 7. \\(\frac{4^{-1}}{-7^0}\\) 8. \\(\frac{3^{-1}}{(-5)^0}\\)

in exercises 9–23, simplify the expression. write your answer using only positive exponents.

1. \\(z^0\\) 10. \\(a^{-8}\\) 11. \\(6a^0b^{-3}\\)
2. \\(14m^{-4}n^0\\) 13. \\(\frac{3^{-2}r^{-4}}{s^0}\\) 14. \\(\frac{2^3a^{-3}}{8^{-1}b^{-5}c^0}\\)
3. \\(\frac{3^5}{3^3}\\) 16. \\(\frac{(-2)^7}{(-2)^5}\\) 17. \\((-5)^3 \cdot (-5)^3\\)
4. \\((q^5)^3\\) 19. \\((a^{-4})^2\\) 20. \\(\frac{c^4 \cdot c^3}{c^9}\\)
5. \\((-4d)^4\\) 22. \\((-3f)^{-3}\\) 23. \\(\left(\frac{4}{x}\

ight)^{-3}\\)

1. a rectangular prism has length \\(x\\), width \\(\frac{x}{2}\\), and height \\(\frac{x}{3}\\). which of the expressions represent the volume of the prism? select all that apply.

a. \\(6^{-1}x^3\\) b. \\(6^{-1}x^{-3}\\) c. \\((6x^{-3})^{-1}\\) d. \\(2^{-1} \cdot 3^{-1} \cdot x^3\\)

Explanation:

To solve the problem of finding the volume of the rectangular prism and identifying the correct expressions, we follow these steps:

Step 1: Recall the formula for the volume of a rectangular prism

The volume \( V \) of a rectangular prism is given by the product of its length, width, and height:
\[ V = \text{length} \times \text{width} \times \text{height} \]

Step 2: Substitute the given dimensions

The length is \( x \), the width is \( \frac{x}{2} \), and the height is \( \frac{x}{3} \). Substituting these into the formula:
\[ V = x \times \frac{x}{2} \times \frac{x}{3} \]

Step 3: Simplify the expression

First, multiply the coefficients and the variables separately:

• For the coefficients: \( 1 \times \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \)
• For the variables: \( x \times x \times x = x^3 \) (using the property \( a^m \cdot a^n = a^{m+n} \), so \( x^1 \cdot x^1 \cdot x^1 = x^{1+1+1} = x^3 \))

Thus, the volume simplifies to:
\[ V = \frac{1}{6} x^3 \]

Step 4: Rewrite \( \frac{1}{6} \) using negative exponents or reciprocal properties

Recall that \( \frac{1}{n} = n^{-1} \) and \( (ab)^{-1} = a^{-1}b^{-1} \).

• \( \frac{1}{6} = 6^{-1} \), so \( \frac{1}{6}x^3 = 6^{-1}x^3 \) (matches option A).
• \( \frac{1}{6} = \frac{1}{2 \times 3} = 2^{-1} \cdot 3^{-1} \) (since \( \frac{1}{2} = 2^{-1} \) and \( \frac{1}{3} = 3^{-1} \)), so \( \frac{1}{6}x^3 = 2^{-1} \cdot 3^{-1} \cdot x^3 \) (matches option D).
• For option C: \( (6x^{-3})^{-1} \). Using the property \( (ab)^n = a^n b^n \) and \( (a^m)^n = a^{mn} \):

\[ (6x^{-3})^{-1} = 6^{-1} \cdot (x^{-3})^{-1} = 6^{-1}x^{3} \] (matches the volume expression).

Step 5: Eliminate incorrect options

• Option B: \( 6^{-1}x^{-3} \) has \( x^{-3} \), which is not equal to \( x^3 \). Eliminate B.

Final Answer

The correct options are:
A. \( 6^{-1}x^3 \)
C. \( (6x^{-3})^{-1} \)
D. \( 2^{-1} \cdot 3^{-1} \cdot x^3 \)

(Note: If we re-express \( (6x^{-3})^{-1} \):
\[ (6x^{-3})^{-1} = 6^{-1} \cdot (x^{-3})^{-1} = 6^{-1}x^{3} \], which matches the volume. Thus, C is also correct.)

Final Answer (Selected Options)

A. \( 6^{-1}x^3 \)
C. \( (6x^{-3})^{-1} \)
D. \( 2^{-1} \cdot 3^{-1} \cdot x^3 \)

Answer:

To solve the problem of finding the volume of the rectangular prism and identifying the correct expressions, we follow these steps:

Step 1: Recall the formula for the volume of a rectangular prism

The volume \( V \) of a rectangular prism is given by the product of its length, width, and height:
\[ V = \text{length} \times \text{width} \times \text{height} \]

Step 2: Substitute the given dimensions

The length is \( x \), the width is \( \frac{x}{2} \), and the height is \( \frac{x}{3} \). Substituting these into the formula:
\[ V = x \times \frac{x}{2} \times \frac{x}{3} \]

Step 3: Simplify the expression

First, multiply the coefficients and the variables separately:

• For the coefficients: \( 1 \times \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \)
• For the variables: \( x \times x \times x = x^3 \) (using the property \( a^m \cdot a^n = a^{m+n} \), so \( x^1 \cdot x^1 \cdot x^1 = x^{1+1+1} = x^3 \))

Thus, the volume simplifies to:
\[ V = \frac{1}{6} x^3 \]

Step 4: Rewrite \( \frac{1}{6} \) using negative exponents or reciprocal properties

Recall that \( \frac{1}{n} = n^{-1} \) and \( (ab)^{-1} = a^{-1}b^{-1} \).

• \( \frac{1}{6} = 6^{-1} \), so \( \frac{1}{6}x^3 = 6^{-1}x^3 \) (matches option A).
• \( \frac{1}{6} = \frac{1}{2 \times 3} = 2^{-1} \cdot 3^{-1} \) (since \( \frac{1}{2} = 2^{-1} \) and \( \frac{1}{3} = 3^{-1} \)), so \( \frac{1}{6}x^3 = 2^{-1} \cdot 3^{-1} \cdot x^3 \) (matches option D).
• For option C: \( (6x^{-3})^{-1} \). Using the property \( (ab)^n = a^n b^n \) and \( (a^m)^n = a^{mn} \):

\[ (6x^{-3})^{-1} = 6^{-1} \cdot (x^{-3})^{-1} = 6^{-1}x^{3} \] (matches the volume expression).

Step 5: Eliminate incorrect options

• Option B: \( 6^{-1}x^{-3} \) has \( x^{-3} \), which is not equal to \( x^3 \). Eliminate B.

Final Answer

The correct options are:
A. \( 6^{-1}x^3 \)
C. \( (6x^{-3})^{-1} \)
D. \( 2^{-1} \cdot 3^{-1} \cdot x^3 \)

(Note: If we re-express \( (6x^{-3})^{-1} \):
\[ (6x^{-3})^{-1} = 6^{-1} \cdot (x^{-3})^{-1} = 6^{-1}x^{3} \], which matches the volume. Thus, C is also correct.)

Final Answer (Selected Options)

A. \( 6^{-1}x^3 \)
C. \( (6x^{-3})^{-1} \)
D. \( 2^{-1} \cdot 3^{-1} \cdot x^3 \)