https://file.aiquickdraw.com/mathbot/file/1727669856953swa5n9bb.jpg

The question asks us to express \(64 = a^b\) in four different ways, where \(a\) and \(b\) are positive integers. We need to find all such pairs of \(a\) and \(b\), and then determine the sum of all possible values of \(a\).

### Step 1: Find all representations of \(64 = a^b\)

We know that \(64 = 2^6\). Let's break down all the possible ways to express 64 as \(a^b\):

1. \(64 = 64^1\) → Here, \(a = 64\) and \(b = 1\).
2. \(64 = 8^2\) → Here, \(a = 8\) and \(b = 2\), since \(8^2 = (2^3)^2 = 2^6\).
3. \(64 = 4^3\) → Here, \(a = 4\) and \(b = 3\), since \(4^3 = (2^2)^3 = 2^6\).
4. \(64 = 2^6\) → Here, \(a = 2\) and \(b = 6\).

### Step 2: Sum of all values of \(a\)
The possible values of \(a\) are \(64, 8, 4,\) and \(2\). 

So, the sum of these values is:
\[
64 + 8 + 4 + 2 = 78
\]

### Final Answer: 
The sum of all possible values of \(a\) is **78**.

Would you like further details on how I derived these steps?

Here are some related questions:
1. What other integers can be expressed as powers in multiple ways like 64?
2. Can the values of \(b\) for such equations ever be non-integers?
3. How would the solution change if negative integers were allowed for \(a\)?
4. What if the question asked for real numbers instead of positive integers?
5. How does the prime factorization of numbers help in solving these types of problems?

**Tip**: When dealing with powers, always check if the base can be broken down into smaller powers of prime numbers for better clarity!

We can write 64 = a^b in four different ways, where a and b are positive integers. What is the sum of all possible values of a?

Learn how to express the number 64 in four different exponential forms with positive integers a and b. Discover all possible values of a and their sum, using concepts of prime factorization and exponents. Ideal for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation