How to multiply and divide terms with the same base

We can multiply powers of the same base even when their exponents are different. Let’s count the factors that are being multiplied, then see what happens when some of those factors divide out.

Multiply the coefficients and add the exponents

In $4 x^{2} \cdot 3 x^{5}$ , multiply the coefficients $4$ and $3$ . The powers have the same base, so there are two factors of $x$ from the first power and five from the second. Add the exponents to count all seven factors.

\begin{array}{l} 4 x^{2} \cdot 3 x^{5} = (4 \cdot 3) x^{2 + 5} \\ = 12 x^{7} \end{array}

If the bases are different, keep their powers separate. For example, $x^{2} \cdot y^{3} = x^{2} y^{3}$ . Adding or subtracting terms has a different rule: to combine those terms, their variable parts, including their exponents, need to match.

Divide the coefficients and subtract the exponents

In $\frac{4 x^{3}}{2 x}$ , divide $4$ by $2$ . One factor of $x$ divides out from the three in the numerator, leaving two factors. We can only divide by $x$ when $x \neq 0$ .

\begin{array}{l} \frac{4 x^{3}}{2 x} = \frac{4}{2} x^{3 - 1} \\ = 2 x^{2} \end{array}

Example: Cancel a factor before combining like terms

\frac{x (x + 2 x - a + b)}{x}

The numerator has $x$ multiplying the whole expression in parentheses. For $x \neq 0$ , divide that outside factor by the $x$ in the denominator, then combine $x + 2 x$ .

\begin{array}{l} x + 2 x - a + b \\ = 3 x - a + b \end{array}

Example: Factor a constant out of a numerator

\frac{(2 x - 4) (x^{2} - x + 3)}{2}

The factor $2 x - 4$ can be written as $2 (x - 2)$ . That makes $2$ a factor of the whole numerator, so it divides out before we multiply the remaining factors.

\begin{array}{l} (x - 2) (x^{2} - x + 3) \\ = x^{3} - x^{2} + 3 x - 2 x^{2} + 2 x - 6 \\ = x^{3} - 3 x^{2} + 5 x - 6 \end{array}

Learn mathKrista KingDecember 15, 2020math, learn online, online course, online math, algebra, algebra i, algebra 1, like terms, combining like terms, multiplying like terms, dividing like terms

How to multiply and divide terms with the same base

Multiply the coefficients and add the exponents

Divide the coefficients and subtract the exponents

Example: Cancel a factor before combining like terms

Example: Factor a constant out of a numerator

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