Algebra basics
Exponent product rules
ALEKS placement
Same base: add exponents when multiplying, subtract when dividing.
What this covers
xaxb=xa+b,xbxa=xa−b,(xa)b=xab- An exponent is a repeat count, not a multiplier: x⁴ means x·x·x·x, four copies multiplied together. That is why x⁴ is 16 when x is 2, not 8.
- The base is the thing being repeated and the exponent is the small raised number counting the copies. These rules only apply between matching bases.
- Confirm the bases match before touching the exponents. Different bases have nothing to combine.
- Why adding works: x³ times x⁴ is three copies of x times four more copies, seven copies in all.
- Why subtracting works: in x⁵ over x², two pairs of x cancel the way a fraction reduces, leaving three on top.
- Handle coefficients separately from the variable parts.
- A power raised to a power multiplies exponents: (x²)³ is three copies of x², six x's in all.
- A product inside parentheses raises every factor.
- Check with a number: at x = 2, x³·x⁴ is 8·16 and x⁷ is 128. Both come out to 128.
Worked example
Worked example
Simplify (3x²y)³ · x⁴.
- 27x6y3⋅x4Raise every factor: 3³ = 27 and (x²)³ = x⁶.
- 27x10y3Same base x, so add 6 + 4.
Another worked example
For x ≠ 0, simplify x³ · x⁴ / x².
- x2x3x4=x3+4−2Join factors in the numerator, then cancel two with the nonzero denominator.
- x5,x=0The original denominator excludes zero even though x⁵ by itself would permit it.
A common mistake
A mistake Lemma catches
From 27x6y3⋅x4
27x24y327x10y3
“You multiplied the exponents 6 and 4, but x^6 · x^4 is 6 factors of x times 4 more — multiplying powers of the same base adds the exponents.”
Try one
Sample problem
Simplify.
x5⋅x2Practice exponent rules free
No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.
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