https://file.aiquickdraw.com/mathbot/file/1726212025642r3fn80e5.jpg

Let's go through the questions one by one and solve them.

### Problem 6:

We need to find a number \( x \) such that:

\[
\frac{\left( -\frac{2}{3} \right)^{-3}}{x} = \left( \frac{9}{8} \right)^{-2}
\]

#### Steps:
1. Simplify both sides.
   \[
   \left( -\frac{2}{3} \right)^{-3} = \left( -\frac{3}{2} \right)^3 = -\frac{27}{8}
   \]
   And:
   \[
   \left( \frac{9}{8} \right)^{-2} = \left( \frac{8}{9} \right)^2 = \frac{64}{81}
   \]

2. Now substitute into the equation:
   \[
   \frac{-\frac{27}{8}}{x} = \frac{64}{81}
   \]

3. Solve for \( x \):
   \[
   x = \frac{-\frac{27}{8}}{\frac{64}{81}} = -\frac{27}{8} \times \frac{81}{64} = -\frac{2187}{512}
   \]

Thus, the number \( x \) is \( -\frac{2187}{512} \).

---

### Problem 7:

We are asked to find the product of 144 multiplied by \( 2^{-3} \), \( 12^{-1} \), and \( 3^{-2} \).

#### Steps:
1. Start with 144.
2. Multiply by \( 2^{-3} \), which is \( \frac{1}{2^3} = \frac{1}{8} \).
3. Multiply by \( 12^{-1} \), which is \( \frac{1}{12} \).
4. Multiply by \( 3^{-2} \), which is \( \frac{1}{3^2} = \frac{1}{9} \).

Now compute:
\[
144 \times \frac{1}{8} \times \frac{1}{12} \times \frac{1}{9} = \frac{144}{8 \times 12 \times 9} = \frac{144}{864} = \frac{1}{6}
\]

Thus, the final product is \( \frac{1}{6} \).

---

### Problem 8:

We need to find the number \( y \) such that:

\[
\frac{\left( \frac{3}{4} \right)^{-4} \times \left( -\frac{2}{3} \right)^2}{y} = \left( \frac{3}{5} \right)^{-2}
\]

#### Steps:
1. Simplify the expressions.
   \[
   \left( \frac{3}{4} \right)^{-4} = \left( \frac{4}{3} \right)^4 = \frac{256}{81}
   \]
   \[
   \left( -\frac{2}{3} \right)^2 = \frac{4}{9}
   \]
   So the product is:
   \[
   \frac{256}{81} \times \frac{4}{9} = \frac{1024}{729}
   \]
   On the right-hand side:
   \[
   \left( \frac{3}{5} \right)^{-2} = \left( \frac{5}{3} \right)^2 = \frac{25}{9}
   \]

2. Substitute into the equation:
   \[
   \frac{\frac{1024}{729}}{y} = \frac{25}{9}
   \]

3. Solve for \( y \):
   \[
   y = \frac{\frac{1024}{729}}{\frac{25}{9}} = \frac{1024}{729} \times \frac{9}{25} = \frac{9216}{18225}
   \]

Thus, the number \( y \) is \( \frac{9216}{18225} \).

---

Let me know if you need further details or have any questions! Here are some related questions:

1. What is the general rule for negative exponents?
2. How do you simplify fractions raised to a negative power?
3. How do you handle division of fractions?
4. Can you explain the reciprocal relationship in fraction exponents?
5. How does raising a negative fraction to an even power affect the result?

**Tip**: When working with exponents and fractions, simplifying both the base and exponent first can make the calculations easier!

By what number should (-2/3)^-3 be divided so that the quotient is (9/8)^-2? The number 144 is multiplied with 2^-3, 12^-1, and 3^-2 consecutively. Find the final product. The product of (3/4)^-4 and (-2/3)^2 is divided by a number so that the quotient is (3/5)^-2. Find the number.

Learn how to solve complex problems involving fractional powers and division of exponents. Includes step-by-step solutions to questions on dividing by negative exponents, multiplying with powers, and simplifying fractional expressions. Ideal for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation