Write the exponential value of -216 by 343 with steps

To express the exponential value of \(-216\) by \(343\), we first note that this is asking for the **division** of \(-216\) by \(343\) in exponential form, typically involving powers or roots.

However, if you're asking for the ratio of \(-216\) to \(343\) as a base raised to an exponent, we can follow these steps:

### Step 1: Prime Factorization of Both Numbers
#### Prime Factorization of \(-216\):
\[
-216 = -6^3
\]
\(6 = 2 \times 3\), so:
\[
-216 = -(2 \times 3)^3 = -2^3 \times 3^3
\]

#### Prime Factorization of \(343\):
\[
343 = 7^3
\]

### Step 2: Write as a Fraction
We are looking at \(\frac{-216}{343}\), which becomes:
\[
\frac{-2^3 \times 3^3}{7^3}
\]

### Step 3: Combine the Powers
Since all the terms are cubed, we can express the result as:
\[
\left(\frac{-2 \times 3}{7}\right)^3 = \left(\frac{-6}{7}\right)^3
\]

Thus, the exponential value of \(-216\) by \(343\) is:
\[
\left(\frac{-6}{7}\right)^3
\]

This means that \(\frac{-216}{343}\) can be expressed as \(\left(\frac{-6}{7}\right)\) raised to the power of \(3\).

---

Would you like more details or have any questions?

### Related Questions:
1. What is the significance of prime factorization in simplifying fractions?
2. How can fractional exponents be simplified?
3. How do you handle negative bases in exponential expressions?
4. Can you convert non-cubic values into similar exponential forms?
5. How can exponential forms be used to simplify large division problems?

### Tip:
Always factorize numbers before applying exponential rules; it simplifies complex calculations significantly.

Learn how to express the exponential value of -216 by 343 with a step-by-step guide. This problem involves prime factorization, fractional powers, and exponent properties, suitable for Grades 9-10 math students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation