Given that $x^a = 4$ and $x^b = 9$, where $a$ and $b$ are positive integers, find the value of $x^{a+b}$.
Understand the Problem
The question states $x^a = 4$ and $x^b = 9$, where $a$ and $b$ are positive integers. We need to find the value of $x^{a+b}$. This can be solved using exponent rules.
Answer
$x^{a+b} = 36$
Answer for screen readers
$x^{a+b} = 36$
Steps to Solve
- Use the exponent rule $x^{a+b} = x^a \cdot x^b$
We can rewrite $x^{a+b}$ as a product of $x^a$ and $x^b$ using the exponent rule that states $x^{a+b} = x^a \cdot x^b$
- Substitute given values
We are given that $x^a = 4$ and $x^b = 9$. Substitute these values into the expression $x^a \cdot x^b$.
$x^{a+b} = x^a \cdot x^b = 4 \cdot 9$
- Calculate the final value
Multiply the values to find the final answer
$4 \cdot 9 = 36$
$x^{a+b} = 36$
More Information
The problem leverages a fundamental property of exponents to simplify the calculation.
Tips
A common mistake would be to try and solve for $x$, $a$, and $b$ individually. This is not necessary and makes the problem much harder. Recognizing and applying the exponent rule directly is the most efficient approach.
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