The expression 5 5 45 6 3 x · 8 82 x is equivalent to b ax ,
where a and b are positive constants and x > 1.
What is the value of a + b ?

The given expression is \( 5^5 \times 45 \times 6^3 \times x \cdot 8 \times 82 \times x \). We are told that this is equivalent to \( b \cdot a^x \), where \( a \) and \( b \) are positive constants, and \( x > 1 \).

### Step-by-step solution:

1. **Group constants together**: Start by separating the constants and the \(x\) terms from the expression:

   \[
   5^5 \times 45 \times 6^3 \times 8 \times 82 \times x^2.
   \]

2. **Calculate the constants**: Now, let's compute the product of the constants:

   - \( 5^5 = 3125 \),
   - \( 6^3 = 216 \),
   - \( 8 = 8 \),
   - \( 45 = 45 \),
   - \( 82 = 82 \).

   Therefore, the product of the constants is:

   \[
   3125 \times 45 \times 216 \times 8 \times 82.
   \]

3. **Simplify the constants product**: Perform the multiplication step-by-step:

   \[
   3125 \times 45 = 140625,
   \]
   \[
   140625 \times 216 = 30375000,
   \]
   \[
   30375000 \times 8 = 243000000,
   \]
   \[
   243000000 \times 82 = 19926000000.
   \]

   Thus, the product of the constants is \( 19926000000 \).

4. **Rearrange the expression**: The expression can now be written as:

   \[
   19926000000 \times x^2.
   \]

5. **Compare with \( b \cdot a^x \)**: Comparing this with \( b \cdot a^x \), we can see that:
   - \( b = 19926000000 \),
   - \( a = x^2 \), which means \( a = x \).

6. **Calculate \( a + b \)**: Now, \( a = x \) and \( b = 19926000000 \). Therefore:

   \[
   a + b = x + 19926000000.
   \]

Since no specific value of \( x \) is given, the result depends on \( x \). However, the expression simplifies to \( a + b = x + 19926000000 \).

---

Would you like to dive deeper into any of the steps or have further clarifications?

Here are five related questions to expand your understanding:
1. How can we apply this method to a similar problem with different constants?
2. What is the significance of separating the constants from the variables in an expression?
3. Can you derive \( a + b \) if \( x = 2 \) in this case?
4. How can you simplify the process of multiplying large constants without a calculator?
5. How would the expression change if there were more variable terms like \( x^3 \)?

**Tip**: When handling large constants, breaking the multiplication into smaller, step-by-step chunks often helps prevent mistakes!

The expression 5^5 × 45 × 6^3 × x · 8 × 82 × x is equivalent to b · a^x, where a and b are positive constants and x > 1. What is the value of a + b?

Explore how to simplify the exponent expression 5^5 × 45 × 6^3 × x · 8 × 82 × x into the form b · a^x, and find the sum of a + b. This solution walks through exponent rules, algebraic manipulation, and the final calculation. Suitable for high school students (Grades 10-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation