LEMMA

Algebra basics

Negative exponents

ALEKS placement

A negative exponent is a reciprocal instruction, not a negative number.

What this covers

x−n=xn1​,x−n1​=xn,x0=1
  • A negative exponent counts divisions, not a minus sign. x⁻³ means 1 divided by x three times, which is 1 over x³.
  • Move a factor across the fraction bar and the sign of its exponent flips. Top to bottom or bottom to top, same rule.
  • Simplify the exponents first. After dividing, write the variable on top, then move it below the bar if its exponent came out negative.
  • Anything nonzero raised to the zero power is 1.
  • Coefficients do not move — only the factors carrying the negative exponent. 2x⁻³ is 2 over x³.
  • Check with numbers: 2⁻³ is 1 ÷ 2 ÷ 2 ÷ 2, which is 0.125, and 1 over 8 is also 0.125.

Worked example

Worked example

Write 4x⁻³y² / (2x y⁻¹) with positive exponents.

  1. 2x−3−1y2−(−1),x=0, y=0Divide coefficients and subtract exponents on matching bases. The original negative powers need x and y nonzero.
  2. 2x−4y3,y=0Simplify the exponents.
  3. x42y3​,y=0Move x⁻⁴ to the denominator to make its exponent positive.
Another worked example

Evaluate 2⁻³ and (−2)⁻³.

  1. 2−3=231​=81​The negative exponent takes a reciprocal of the positive base’s power.
  2. (−2)−3=(−2)31​=−81​The negative base is raised to an odd power; that supplies the negative sign.

A common mistake

A mistake Lemma catches

From 2x−4y3

2x−4y3=2x4y3​2x−4y3=x42y3​

“The negative exponent belongs to x alone, so only x⁴ moves under the fraction bar: 2x⁻⁴y³ = 2 · (1/x⁴) · y³ = 2y³/x⁴. The 2 has no exponent and stays on top; putting it underneath divides by 2 instead of multiplying.”

Try one

Sample problem

Rewrite using positive exponents only.

x−3

Practice negative powers free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

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