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8th Grade · 8.EE.A.1

Exponent rules in 8th grade

Exponent rules are the shortcuts for working with powers such as 232^3 or 5−25^{-2}. In 232^3, the small raised number 3 is the exponent, and it says how many copies of the base, 2, to multiply: 23=2×2×2=82^3 = 2 \times 2 \times 2 = 8. In 8th grade, your child learns the properties of exponents, including zero and negative exponents, and uses them to simplify expressions.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

8.EE.A.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions.

What it means, in plain words

A power has two parts. The base is the number that gets multiplied, and the exponent says how many copies of it to multiply. In 232^3, the base is 2 and the exponent is 3. All the exponent rules come from counting those copies. A child who forgets a rule can write the factors out and find it again.

To multiply powers with the same base, add the exponents: 23×24=272^3 \times 2^4 = 2^7, because three 2s and four 2s make seven 2s. To divide, subtract them: 56÷52=545^6 \div 5^2 = 5^4. A power of a power multiplies the exponents, so (32)4=38(3^2)^4 = 3^8. When a product is raised to a power, each factor gets the exponent: (2⋅5)2=22⋅52(2 \cdot 5)^2 = 2^2 \cdot 5^2. Textbooks also call these the laws of exponents.

The exponent rules in 8th grade add two new cases. Any number except 0 raised to the power 0 equals 1, so 70=17^0 = 1. A negative exponent means one over the power, so 3−2=132=193^{-2} = \frac{1}{3^2} = \frac{1}{9}. A negative exponent never makes the number itself negative. Both cases follow from the division rule: 52÷525^2 \div 5^2 is 1, and the rule turns it into 505^0.

Your child should be able to:

  • Multiply and divide powers with the same base by adding or subtracting the exponents
  • Raise a power to a power and a product to a power
  • Explain why 70=17^0 = 1 and rewrite a negative exponent as a fraction
  • Simplify an expression such as 32×3−53^2 \times 3^{-5} to a single number

Example

Simplify 32×3−53^2 \times 3^{-5} and write the answer as a fraction.

  1. Both powers have base 3, so add the exponents: 2+(−5)=−32 + (-5) = -3.
  2. 32×3−5=3−33^2 \times 3^{-5} = 3^{-3}
  3. A negative exponent means one over the power: 3−3=133=1273^{-3} = \frac{1}{3^3} = \frac{1}{27}.
  4. The answer is one twenty-seventh, 127\frac{1}{27}.

Example

Write 252−1\frac{2^5}{2^{-1}} as a single power of 2 and find its value.

  1. Both powers have base 2, so subtract the exponents: 5−(−1)=65 - (-1) = 6.
  2. 252−1=26\frac{2^5}{2^{-1}} = 2^6
  3. 26=2×2×2×2×2×2=642^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64
  4. The answer is 2 to the sixth power, which equals 64.

Where children usually get stuck

  • Your child writes 23×24=472^3 \times 2^4 = 4^7 or 23×24=2122^3 \times 2^4 = 2^{12}.

    The rules got mixed up. The bases were multiplied along with the exponents, or the exponents were multiplied instead of added. The rule was learned as a pattern of symbols, not as counting factors.

    What helps. Ask your child to write the factors out once: 2×2×22 \times 2 \times 2 times 2×2×2×22 \times 2 \times 2 \times 2. Counting seven 2s shows that the base stays 2 and the exponents add.

  • Your child writes 2−1=−22^{-1} = -2 or 3−2=−93^{-2} = -9.

    The minus sign in the exponent looks like the minus sign of a negative number. Nothing in the symbol says "one over".

    What helps. Write a column of powers: 23=82^3 = 8, 22=42^2 = 4, 21=22^1 = 2, 20=12^0 = 1. Each step down divides by 2. Ask what comes next. The pattern gives 2−1=122^{-1} = \frac{1}{2} and 2−2=142^{-2} = \frac{1}{4}.

  • Your child writes 50=05^0 = 0.

    Raising to the power 0 sounds like multiplying by 0, so 0 feels like the natural answer.

    What helps. Use division. 53÷535^3 \div 5^3 is 125 divided by 125, which is 1. The division rule turns the same problem into 53−3=505^{3-3} = 5^0, so 505^0 has to be 1.

Try four practice questions

Real 8th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

  1. 1Warm-up
    Which expression is equivalent to the one shown?
    (2⋅5)2\displaystyle \left(2\cdot 5\right)^2
    A
    22⋅53\displaystyle 2^2 \cdot 5^3
    B
    22⋅52\displaystyle 2^2 \cdot 5^2
    C
    2⋅52\displaystyle 2 \cdot 5^2
    D
    22+\displaystyle 2^2 + {}
    52\displaystyle 5^2
    Show the answer and how to solve it

    The answer is 22⋅522^{2} \cdot 5^{2} because the outside exponent 22 applies to both factors in the product 2⋅52 \cdot 5.

    1. 1Look at the expression. The exponent 22 is on the entire product 2⋅52 \cdot 5, not on just one factor.
    2. 2The product 2⋅52 \cdot 5 is multiplied by itself 22 times. Regrouping gives 22 factors of 22 and 22 factors of 55.
    3. 3So the matching expression is 22⋅522^{2} \cdot 5^{2}.

    Common mistake. A common mistake is giving the exponent to only one factor, such as 2⋅522 \cdot 5^{2}, or turning the product into a sum, such as 22+522^{2} + 5^{2}. Both factors must keep multiplication and both get the same outside exponent.

    The idea underneath. The expression (a⋅b)n(a \cdot b)^{n} means nn copies of a⋅ba \cdot b multiplied together, so there end up being nn copies of each factor.

  2. 2On level
    Rewrite the expression below as a product of powers. Do not evaluate.
    (3⋅8)4\displaystyle \left(3 \cdot 8\right)^{4}
    Loading...
    Show the answer and how to solve it

    Answer. 3^{4} \cdot 8^{4}

    The answer is 34⋅843^{4} \cdot 8^{4} because the exponent 4 goes on both factors, 3 and 8.

    1. 1Look at the expression. The whole product 3⋅83 \cdot 8 is inside the parentheses, and the exponent is 4.
    2. 2The outside exponent 4 applies to each factor in the product.
    3. 3Write each factor with exponent 4.
    4. 4Stop there. The problem asks for a product of powers, so you do not multiply 3 and 8 or evaluate the powers.

    Common mistake. A common mistake is writing 3⋅843 \cdot 8^{4} or 34⋅83^{4} \cdot 8. The outside exponent must be applied to both original factors.

    The idea underneath. An exponent on a product means the whole product is repeated that many times, so each original factor is repeated that many times too.

  3. 3On level
    Fill in the missing exponents to make the simplification chain correct.
    a−1⋅(a4)2÷a\displaystyle a^{-1} \cdot (a^4)^2 \div a
    Step 1
    =a−1⋅a\displaystyle = a^{-1} \cdot a
    ÷a\displaystyle \div a
    Step 2
    =a\displaystyle = a
    Show the answer and how to solve it

    The missing exponents are 88 and 66 because (a4)2(a^{4})^{2} becomes a8a^{8}, and then −1+8−1=6-1 + 8 - 1 = 6.

    1. 1Start with the nested power (a4)2(a^{4})^{2}. Multiply the exponents.
    2. 2Put that result back into the expression. The divisor aa is a bare base, so it has exponent 11.
    3. 3Add the exponents in the numerator and subtract the divisor exponent.

    Common mistake. A common mistake is adding 44 and 22 to get 66 in the first blank instead of multiplying them to get 88. Another is forgetting that dividing by the bare base aa means subtracting 11 from the total exponent, writing −1+8=7-1 + 8 = 7 instead of 66.

    The idea underneath. When every factor has the same base, the exponents track how many copies of that base you have: a power of a power multiplies copies, multiplication adds copies, and division removes copies.

  4. 4Stretch
    Look at the quotient below. Which single power is equivalent to it?
    545−3\displaystyle \frac{5^{4}}{5^{-3}}
    A
    57\displaystyle 5^{7}
    B
    51\displaystyle 5^{1}
    C
    5−7\displaystyle 5^{-7}
    D
    54\displaystyle 5^{4}
    Show the answer and how to solve it

    The answer is 575^{7} because dividing same-base powers means subtracting exponents: 4−(−3)=74 - (-3) = 7.

    1. 1Look at the quotient. Both powers share the same base, 5. The denominator exponent is −3-3.
    2. 2Apply the quotient rule: subtract the denominator exponent from the numerator exponent. Write the denominator exponent in parentheses to keep track of its sign.
    3. 3Subtracting a negative is the same as adding: 4−(−3)=4+34 - (-3) = 4 + 3.
    4. 4Compute 4+3=74 + 3 = 7. The equivalent single power is 575^{7}.

    Common mistake. A common mistake is choosing 515^{1} by computing 4−3=14 - 3 = 1, which ignores the negative sign on the denominator exponent. The denominator exponent is −3-3, so the correct subtraction is 4−(−3)=74 - (-3) = 7.

    The idea underneath. Subtracting exponents works because equal factors of the base cancel; a negative exponent means more factors remain in the denominator than in the numerator.