https://file.aiquickdraw.com/mathbot/file/1730275967228jwq2lt5l.jpg

To solve the equation given in the image:

\[
(5^{\frac{x}{3}})(5^{\frac{x}{5}}) = 5^4
\]

we can use properties of exponents. When multiplying terms with the same base, we add the exponents. Therefore:

\[
5^{\frac{x}{3} + \frac{x}{5}} = 5^4
\]

Now, since the bases are the same, we can equate the exponents:

\[
\frac{x}{3} + \frac{x}{5} = 4
\]

To solve for \( x \), we need to combine the terms on the left-hand side. First, find a common denominator for \(\frac{x}{3}\) and \(\frac{x}{5}\), which is 15:

\[
\frac{5x}{15} + \frac{3x}{15} = 4
\]

Combine the fractions:

\[
\frac{8x}{15} = 4
\]

Now, solve for \( x \) by multiplying both sides by \(\frac{15}{8}\):

\[
x = 4 \cdot \frac{15}{8} = \frac{60}{8} = \frac{15}{2}
\]

So, the solution is:

\[
x = \frac{15}{2}
\]

Would you like further clarification on any step? 

Here are related questions that could enhance understanding:

1. How can you solve exponential equations with different bases?
2. What other exponent properties are useful in simplifying equations?
3. How would the solution change if the base were different?
4. Can you solve similar problems without simplifying the exponent?
5. What is the significance of finding a common denominator in this context?

**Tip:** When dealing with exponents, always look for ways to simplify using properties like addition, subtraction, or division when the bases are the same.

Solve the equation \((5^{\frac{x}{3}})(5^{\frac{x}{5}}) = 5^4\).

Learn how to solve the exponential equation (5^{x/3})(5^{x/5}) = 5^4 using properties of exponents. This problem involves adding exponents and equating expressions with the same base. Suitable for high school students in 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