Algebra 1

Exponent rules and how to use them

Exponent rules are shortcuts for repeated multiplication. When you multiply powers with the same base, add the exponents. When you divide them, subtract the exponents. A power raised to a power multiplies the exponents. Any nonzero number to the zero power is 1, and a negative exponent means a reciprocal, so x−n=1xnx^{-n} = \frac{1}{x^n}. The add and subtract rules only work when the bases match.

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The key idea

An exponent counts how many times a base is multiplied by itself. Every rule comes from counting factors:

x3⋅x4=(x⋅x⋅x)(x⋅x⋅x⋅x)=x7x^3 \cdot x^4 = (x \cdot x \cdot x)(x \cdot x \cdot x \cdot x) = x^{7}
The rules (x and y are not zero)
RuleIn symbolsExample
Productxa⋅xb=xa+bx^a \cdot x^b = x^{a+b}x3⋅x4=x7x^3 \cdot x^4 = x^7
Quotientxaxb=xa−b\frac{x^a}{x^b} = x^{a-b}x9x2=x7\frac{x^9}{x^2} = x^7
Power of a power(xa)b=xab(x^a)^b = x^{ab}(x2)5=x10(x^2)^5 = x^{10}
Power of a product(xy)a=xaya(xy)^a = x^a y^a(3x)2=9x2(3x)^2 = 9x^2
Power of a quotient(xy)a=xaya\left(\frac{x}{y}\right)^a = \frac{x^a}{y^a}(2x)3=8x3\left(\frac{2}{x}\right)^3 = \frac{8}{x^3}
Zero exponentx0=1x^0 = 170=17^0 = 1
Negative exponentx−a=1xax^{-a} = \frac{1}{x^a}2−3=182^{-3} = \frac{1}{8}

Worked examples

Example 1: product rule with coefficients

Problem Simplify (2x3y)(5x2y4)(2x^3y)(5x^2y^4).

  1. Group the numbers, the x's and the y's. Remember that a plain yy is y1y^1.
    (2⋅5)(x3⋅x2)(y1⋅y4)=10x3+2y1+4(2 \cdot 5)(x^3 \cdot x^2)(y^1 \cdot y^4) = 10x^{3+2}y^{1+4}
  2. Multiply the numbers and add the exponents.
    10x3+2y1+4=10x5y510x^{3+2}y^{1+4} = 10x^5y^5

Answer 10x5y510x^5y^5

Example 2: power of a product

Problem Simplify (3a2b4)3(3a^2b^4)^3.

  1. The outside exponent goes to every factor inside, including the 3.
    (3a2b4)3=33(a2)3(b4)3(3a^2b^4)^3 = 3^3(a^2)^3(b^4)^3
  2. Multiply the exponents.
    33(a2)3(b4)3=27a6b123^3(a^2)^3(b^4)^3 = 27a^6b^{12}

Answer 27a6b1227a^6b^{12}

Example 3: quotient with a negative exponent

Problem Simplify 12x5y24x7y−1\frac{12x^5y^2}{4x^7y^{-1}}. Write the answer with positive exponents.

  1. Divide the numbers and subtract exponents for each base (top minus bottom).
    12x5y24x7y−1=3x5−7y2−(−1)\frac{12x^5y^2}{4x^7y^{-1}} = 3x^{5-7}y^{2-(-1)}
  2. Simplify the exponents.
    3x5−7y2−(−1)=3x−2y33x^{5-7}y^{2-(-1)} = 3x^{-2}y^{3}
  3. A negative exponent means the factor belongs on the other side of the fraction bar.
    3x−2y3=3y3x23x^{-2}y^{3} = \frac{3y^3}{x^2}

Answer 3y3x2\frac{3y^3}{x^2}

Example 4 (test-hard): every rule at once

Problem Simplify (2x−2y3)2⋅x58xy4\frac{(2x^{-2}y^3)^2 \cdot x^5}{8xy^4}.

  1. Start inside the top. Square every factor of 2x−2y32x^{-2}y^3.
    (2x−2y3)2=4x−4y6(2x^{-2}y^3)^2 = 4x^{-4}y^{6}
  2. Multiply by x5x^5: add the x exponents, −4+5=1-4 + 5 = 1.
    4x−4y6⋅x5=4xy64x^{-4}y^{6} \cdot x^5 = 4xy^6
  3. Now divide by 8xy48xy^4. The x's cancel (x1−1=x0=1x^{1-1} = x^0 = 1), and 48=12\frac{4}{8} = \frac{1}{2}.
    4xy68xy4=12y2\frac{4xy^6}{8xy^4} = \frac{1}{2}y^{2}

Answer y22\frac{y^2}{2}

Common mistakes

  • Multiplying the bases. 23⋅242^3 \cdot 2^4 is 272^7, not 474^7. The base stays the same; only the exponents combine.
  • Combining different bases. x2⋅y3x^2 \cdot y^3 cannot be simplified. The rules need the same base.
  • Spreading an exponent over a sum. (x+3)2(x + 3)^2 is not x2+9x^2 + 9. The power-of-a-product rule is for multiplication only. See multiplying polynomials.
  • Treating a negative exponent as a negative number. 2−3=182^{-3} = \frac{1}{8}, a small positive number, not −8-8.
  • Forgetting the coefficient. (3x)2=9x2(3x)^2 = 9x^2, but 3x23x^2 means only the x is squared.
  • Saying x0=0x^0 = 0. Any nonzero base to the zero power is 1. So 5x0=55x^0 = 5.

Quick methods

Practice

5 practice questions

  1. Simplify x4⋅x6x^4 \cdot x^6.

    1. x10x^{10}
    2. x24x^{24}
    3. x2x^{2}
    4. 2x102x^{10}
    Show answer

    Answer: x10x^{10}

    Same base, multiplying: add the exponents, 4+6=104 + 6 = 10. Multiplying the exponents (24) is the power-of-a-power rule, which does not apply here.

  2. Simplify (2y3)4(2y^3)^4.

    1. 2y122y^{12}
    2. 8y128y^{12}
    3. 16y716y^{7}
    4. 16y1216y^{12}
    Show answer

    Answer: 16y1216y^{12}

    The 4 applies to the 2 and to y3y^3: 24=162^4 = 16 and (y3)4=y12(y^3)^4 = y^{12}. Choice 2y122y^{12} forgets the coefficient, and 8y128y^{12} multiplies 2⋅42 \cdot 4 instead of raising 2 to the 4th.

  3. What is 5−25^{-2}?

    1. −25-25
    2. −10-10
    3. 125\frac{1}{25}
    4. 110\frac{1}{10}
    Show answer

    Answer: 125\frac{1}{25}

    A negative exponent means reciprocal: 5−2=152=1255^{-2} = \frac{1}{5^2} = \frac{1}{25}. It is never negative just because the exponent is.

  4. Simplify 18a4b−26ab3\frac{18a^4b^{-2}}{6ab^3}.

    1. 3a3b53a^3b^5
    2. 3a3b5\frac{3a^3}{b^5}
    3. 3a4b\frac{3a^4}{b}
    4. 12a3b−512a^3b^{-5}
    Show answer

    Answer: 3a3b5\frac{3a^3}{b^5}

    186=3\frac{18}{6} = 3, a4−1=a3a^{4-1} = a^3, and b−2−3=b−5=1b5b^{-2-3} = b^{-5} = \frac{1}{b^5}. The choice 3a3b53a^3b^5 drops the negative sign, and 12 subtracts the coefficients instead of dividing.

  5. Solve 3x+4=9x3^{x+4} = 9^{x}.

    1. x=−4x = -4
    2. x=2x = 2
    3. x=4x = 4
    4. x=8x = 8
    Show answer

    Answer: x=4x = 4

    Write 9 as 323^2, so 9x=32x9^x = 3^{2x}. With the same base on both sides, the exponents must match: x+4=2xx + 4 = 2x, so x=4x = 4. Check: 38=65613^8 = 6561 and 94=65619^4 = 6561.

Frequently asked questions

Why is any number to the zero power equal to 1?

Use the quotient rule on something divided by itself: x3x3=x3−3=x0\frac{x^3}{x^3} = x^{3-3} = x^0. But anything nonzero divided by itself is 1. So x0x^0 has to be 1 for the rules to agree. (000^0 is left undefined in Algebra 1.)

What does a negative exponent mean?

It means take the reciprocal. x−3=1x3x^{-3} = \frac{1}{x^3}, and 1x−3=x3\frac{1}{x^{-3}} = x^3. It tells you which side of the fraction bar the factor belongs on. It does not make the number negative.

Do the exponent rules work with different bases?

The add and subtract rules do not. 23⋅322^3 \cdot 3^2 cannot be combined into one power. The power-of-a-product rule is the exception: 23⋅33=632^3 \cdot 3^3 = 6^3, because the exponents match even though the bases do not.

What is a fractional exponent?

A fractional exponent is a root. An exponent of one half means square root, so x12=xx^{\frac{1}{2}} = \sqrt{x}, and an exponent of one third means cube root. The same rules still apply. Adding the exponents in x12⋅x12x^{\frac{1}{2}} \cdot x^{\frac{1}{2}} gives x1x^1, which is why a square root times itself gives back the number you started with. Most Algebra 1 classes meet these again in Algebra 2.

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  • “Do I add or multiply the exponents here?”
  • “Why is 2 to the -3 not negative 8?”
  • “Check my steps on this simplify problem, I think I messed up a negative exponent.”

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