Multiplying and dividing terms
Work separately with signed coefficients and the exponents of each base. All original denominator factors must be nonzero
- Coefficients multiply Multiply the signed numerical coefficients \(\displaystyle (-3)\times4=-12\)
- Matching factors join Add exponents only for the same base \(\displaystyle a^{2}\times a^{3}=a^{5}\)
- Quotients cancel factors Subtract denominator exponents from numerator exponents. A divisor must be nonzero \(\displaystyle \frac{x^{5}}{x^{2}}=x^{3}\)
- Different bases stay separate Keep a separate exponent for every pronumeral \(\displaystyle a^{2}\times b^{3}=a^{2}b^{3}\)
- Reduce the coefficient Leave a non-integer coefficient as an exact simplified fraction \(\displaystyle \frac{6a^{3}}{4a}=\frac{3a^{2}}{2}\)
Worked example
Simplify the quotient, keeping the restrictions a nonzero and b nonzero
- \(\displaystyle 18\div6=3\)Divide the numerical coefficients
- \(\displaystyle a^{5}\div a=a^{4}\)Subtract one from five
- \(\displaystyle b^{2}\div b=b\)Subtract one from two
- \(\displaystyle 3a^{4}b\)Combine the result. The original restrictions still apply
Common mistakes
Difficulty levels
Questions about this topic
Why do product exponents add?+
Powers count repeated factors. Multiplication joins the two sets of factors
Do the coefficients add too?+
No. Multiply or divide coefficients according to the displayed operation
What if the bases are different?+
Keep each base and its own exponent. Only matching bases can be combined by the exponent laws
Can I substitute zero after cancelling a denominator factor?+
Only if zero was allowed by the original expression. Cancellation never restores an excluded value
What if more factors occur in the denominator?+
They remain in the denominator and can also be written using negative indices. Later index-law lessons develop that case explicitly
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 3: Algebra and indices