Which expression is equivalent to 3a^3b^4c / 6a^5b for all positive values of a, b, and c?
Understand the Problem
The question is asking for an expression that is equivalent to the given algebraic fraction for all positive values of a, b, and c. This involves simplifying the fraction and possibly finding a common form or expression that matches it.
Answer
The equivalent expression is $$ \frac{b^3c}{2a^2} $$
Answer for screen readers
The equivalent expression is
$$ \frac{b^3c}{2a^2} $$
Steps to Solve
- Identify the Fraction The given algebraic fraction is
$$ \frac{3a^3b^4c}{6a^5b} $$
- Factor Out Common Terms Let's simplify the fraction by factoring out common terms in the numerator and denominator.
- The numerator has $3a^3b^4c$, and the denominator has $6a^5b$.
- Simplify the Numerical Coefficients Divide the numerical coefficients (3 in the numerator and 6 in the denominator):
$$ \frac{3}{6} = \frac{1}{2} $$
- Simplify the Variables Now simplify the variables separately:
- For $a$:
$$ \frac{a^3}{a^5} = a^{3-5} = a^{-2} $$
- For $b$:
$$ \frac{b^4}{b^1} = b^{4-1} = b^{3} $$
- Combine Simplified Terms Now combine all the simplified parts back together. The expression is:
$$ \frac{1}{2} a^{-2} b^3 c $$
Since $a^{-2} = \frac{1}{a^2}$, we can write the expression as:
$$ \frac{b^3c}{2a^2} $$
The equivalent expression is
$$ \frac{b^3c}{2a^2} $$
More Information
This expression retains the same value as the original fraction for all positive values of $a$, $b$, and $c$ because it is obtained by directly simplifying the algebraic fraction.
Tips
- Forgetting to simplify both the numerical and variable parts separately.
- Not paying attention to the rules of exponents, especially with negative exponents.
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