Simplify. Express your answer using positive exponents. $\newline$ $\frac{6f^3}{2f \cdot f^3}$

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Algebra 1

Simplify exponential expressions using the multiplication and division rules

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Q. Simplify. Express your answer using positive exponents.

\newline

\frac{6f^3}{2f \cdot f^3}

Write Expression, Identify Like Terms: Write down the given expression and identify like terms. $\newline$ The given expression is $\frac{6f^3}{2f \cdot f^3}$ . We can see that $f$ appears in both the numerator and the denominator, which means we can simplify by canceling out common factors.
Factor Out Common Terms: Factor out the common terms. $\newline$ We can rewrite the expression by separating the coefficients and the powers of $f$ . $\newline$ $\frac{6f^3}{2f \cdot f^3} = \frac{6}{2} \cdot \frac{f^3}{f^1} \cdot \frac{1}{f^3}$
Simplify Coefficients: Simplify the coefficients. $\newline$ $6$ divided by $2$ is $3$ , so we have: $\newline$ (\frac{$6$}{$2$}) \cdot (\frac{f^$3$}{f^$1$}) \cdot (\frac{$1$}{f^$3$}) = \(3 \cdot (\frac{f^ $3$ }{f^ $1$ }) \cdot (\frac{ $1$ }{f^ $3$ })
Simplify Powers of f: Simplify the powers of f. $\newline$ When dividing powers with the same base, we subtract the exponents. $\newline$ $f^3/f^1 = f^{(3-1)} = f^2$ $\newline$ $1/f^3 = f^{-3}$ $\newline$ So we have: $\newline$ $3 \times (f^3/f^1) \times (1/f^3) = 3 \times f^2 \times f^{-3}$
Combine Powers of f: Combine the powers of f. $\newline$ When multiplying powers with the same base, we add the exponents. $\newline$ $f^2 \cdot f^{-3} = f^{2-3} = f^{-1}$ $\newline$ However, we want to express our answer using positive exponents. $\newline$ $f^{-1}$ is the same as $\frac{1}{f^1}$ or simply $\frac{1}{f}$ . $\newline$ So we have: $\newline$ $3 \cdot f^2 \cdot f^{-3} = 3 \cdot \left(\frac{1}{f}\right)$
Write Final Expression: Write the final simplified expression. $\newline$ The final expression is: $\newline$ $3 \times \left(\frac{1}{f}\right) = \frac{3}{f}$