In a math contest, students are asked to solve a challenging problem involving fractions. One of the problems requires them to simplify the expression (-5/9)^6. The contestants need to calculate the value of (-5/9)^6 to proceed to the next round. Find the simplified value of (-5/9)^6 to help the students advance in the competition. Show all work and write your final answer in a complete sentence with appropriate units.

To solve the expression \(\left(-\frac{5}{9}\right)^6\), we will follow these steps:

### Step 1: Simplify the expression by applying the exponent
The given expression is \(\left(-\frac{5}{9}\right)^6\). When raising a fraction to an exponent, both the numerator and the denominator are raised to that exponent:

\[
\left(-\frac{5}{9}\right)^6 = \frac{(-5)^6}{9^6}
\]

### Step 2: Calculate the powers
1. Calculate \((-5)^6\):
   - Since the exponent is even, \((-5)^6\) will be positive.
   - \( (-5)^6 = (-5) \times (-5) \times (-5) \times (-5) \times (-5) \times (-5) = 15625 \).
   
2. Calculate \(9^6\):
   - \(9^6 = 9 \times 9 \times 9 \times 9 \times 9 \times 9 = 531441\).

### Step 3: Write the result as a fraction
Now, substitute the calculated values into the fraction:

\[
\left(-\frac{5}{9}\right)^6 = \frac{15625}{531441}
\]

### Final Answer
The simplified value of \(\left(-\frac{5}{9}\right)^6\) is \(\frac{15625}{531441}\).

So, the final answer is:
"The simplified value of \(\left(-\frac{5}{9}\right)^6\) is \(\frac{15625}{531441}\)."

Would you like any further details or have any questions?

---

Here are 5 related questions:
1. How would the result change if the exponent were odd instead of even?
2. What is the decimal equivalent of the fraction \(\frac{15625}{531441}\)?
3. How can you simplify the expression \(\left(\frac{5}{9}\right)^6\) without calculating the exact value?
4. What would the answer be if the fraction were \(\left(\frac{5}{9}\right)^{-6}\)?
5. How does the sign of the base affect the outcome when the exponent is even?

**Tip:** When raising a negative number to an even power, the result is always positive.

Learn how to simplify the expression (-5/9)^6 with a detailed, step-by-step explanation. This problem covers key mathematical concepts, including exponents, fraction simplification, and properties of exponents. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation