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6th Grade · 6.EE.A.1

Exponents in 6th grade

Exponents in 6th grade math are the small raised numbers, such as the 3 in 535^3. An exponent says how many copies of a number to multiply together. Your child learns to write repeated multiplication with exponents and to find the value of expressions that contain them.

What the standard says

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

6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents.

What it means, in plain words

In 535^3, the 5 is the base, the number being multiplied. The raised 3 is the exponent, and it says how many copies of the base to multiply: 53=5×5×5=1255^3 = 5 \times 5 \times 5 = 125. The whole thing, 535^3, is called a power. Exponents save space: 7×7×7×77 \times 7 \times 7 \times 7 is written 747^4.

Evaluating exponents means finding the number a power stands for. Children read 525^2 as "5 squared" and 535^3 as "5 cubed" or "5 to the third power". The exponent is not a factor: 535^3 is 125, not 15. In 6th grade the exponents are whole numbers. The base can be a fraction too, so (12)3=18(\frac{1}{2})^3 = \frac{1}{8}.

Expressions with exponents follow the order of operations. Exponents come right after grouping symbols, before multiplying and dividing. So 4+32=4+9=134 + 3^2 = 4 + 9 = 13, but (4+3)2=72=49(4 + 3)^2 = 7^2 = 49. This is the E in PEMDAS.

Your child should be able to:

  • Name the base and the exponent in a power such as 535^3
  • Write repeated multiplication with an exponent, such as 7×7×7×7=747 \times 7 \times 7 \times 7 = 7^4
  • Evaluate a power by multiplying copies of the base
  • Evaluate expressions with exponents using the order of operations

Example

What is the value of 343^4?

  1. The exponent 4 says to multiply four copies of 3: 3×3×3×33 \times 3 \times 3 \times 3.
  2. 3×3=93 \times 3 = 9
  3. 9×3=279 \times 3 = 27
  4. 27×3=8127 \times 3 = 81
  5. So 34=813^4 = 81, not 12.

Example

Evaluate 50−62÷450 - 6^2 \div 4.

  1. Exponent first: 62=6×6=366^2 = 6 \times 6 = 36.
  2. Then divide: 36÷4=936 \div 4 = 9.
  3. Then subtract: 50−9=4150 - 9 = 41.
  4. The value of the expression is 41.

Where children usually get stuck

  • Your child writes 43=124^3 = 12.

    The child multiplied the base by the exponent. The raised number looks like an ordinary factor.

    What helps. Have your child write the copies out every time for a while: 43=4×4×4=644^3 = 4 \times 4 \times 4 = 64. With the copies on paper, 12 looks clearly wrong.

  • Your child treats 343^4 and 434^3 as the same number.

    Swapping the numbers never changes a multiplication such as 3×43 \times 4, so the child expects the same here.

    What helps. Write both out side by side: 34=3×3×3×3=813^4 = 3 \times 3 \times 3 \times 3 = 81 and 43=4×4×4=644^3 = 4 \times 4 \times 4 = 64. The base and the exponent do different jobs.

  • Your child works out (1+4)2(1 + 4)^2 as 1+16=171 + 16 = 17.

    The exponent sits right next to the 4, so it looks as if it belongs only to the 4. The parentheses mean the whole sum is squared, so the answer is 52=255^2 = 25.

    What helps. Ask your child to work out the inside of the parentheses first and write 525^2 on its own line before going on.

Try four practice questions

Real 6th 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
    Fill in the missing numbers to evaluate the expression.
    22+5\displaystyle 2^{2} + 5
    Substitute
    =\displaystyle =
    +5\displaystyle + 5
    Evaluate
    =\displaystyle =
    Show the answer and how to solve it

    The first blank is 4 and the second blank is 9, so the expression equals 9.

    1. 1Start with the power. Evaluate 222^{2} first.
    2. 2Substitute 4 for 222^{2} in the expression.
    3. 3Now add 4 and 5.

    Common mistake. A common mistake is treating 222^{2} as 2×2=42 \times 2 = 4 but then using 2+2=42 + 2 = 4—which happens to be the same here—or, more often, writing 222^{2} as 2×2=42 \times 2 = 4 correctly but then confusing 4+54 + 5 with 4+2=64 + 2 = 6. Remember: substitute first, then add.

    The idea underneath. A power becomes one number first, and then that number is used in the remaining operation.

  2. 2On level
    A student evaluated the expression below.
    Line 1
    3(2+3)2\displaystyle 3(2 + 3)^2
    Line 2
    =3(5)2\displaystyle = 3(5)^2
    Line 3
    =152\displaystyle = 15^2
    Line 4
    =225\displaystyle = 225
    Select the statement that best describes the student's work.
    A
    The number outside the parentheses was multiplied too soon.
    B
    The exponent was used on only one number inside the parentheses.
    C
    The exponent was used on each number inside the parentheses.
    D
    The work is correct.
    Show the answer and how to solve it

    The correct choice is "The number outside the parentheses was multiplied too soon" because the student changed 3(5)23(5)^{2} into 15215^{2} instead of finding 525^{2} first.

    1. 1Look at the student's work line by line. The move from 3(2+3)23(2 + 3)^{2} to 3(5)23(5)^{2} is correct because 2+3=52 + 3 = 5.
    2. 2The error appears on line 3. In 3(5)23(5)^{2}, the 3 is outside the exponent and should not be multiplied by 5 before the square is applied.
    3. 3Square 5 first, then multiply by 3.
    4. 4So the best description is that the number outside the parentheses was multiplied too soon.

    Common mistake. A common mistake is treating 3(5)23(5)^{2} as (3⋅5)2=152(3 \cdot 5)^{2} = 15^{2}. Always apply the exponent to 5 first: 52=255^{2} = 25, then multiply by 3 to get 75.

    The idea underneath. A number written next to parentheses stays outside the power; first evaluate the parentheses and exponent, then multiply by the outside number.

  3. 3On level
    Look at the power below.
    25\displaystyle 2^{5}
    Select its value.
    A
    32
    B
    16
    C
    10
    D
    64
    Show the answer and how to solve it

    The answer is 3232 because 252^{5} means multiply 22 by itself 55 times.

    1. 1The base is 22 and the exponent is 55. That means 22 is used as a factor 55 times.
    2. 2Write the power as repeated multiplication and find the product.
    3. 3So the correct choice is 3232.

    Common mistake. A common mistake is choosing 1010 by multiplying the base and exponent. In 252^{5}, the 55 tells how many 22s to multiply, not what to multiply 22 by.

    The idea underneath. In a power, the base is the repeated factor and the exponent tells how many times that factor appears.

  4. 4Stretch
    Find the value of the power shown below.
    25\displaystyle 2^{5}
    Type the value as a whole number.
    Value:
    Show the answer and how to solve it

    Answer. 32

    The value is 32 because 2 times 2 times 2 times 2 times 2 = 32.

    1. 1The base is 2 and the exponent is 5. That means multiply 2 by itself 5 times.
    2. 2Write the power as repeated multiplication — five factors of 2.
    3. 3Multiply the first two factors.
    4. 4Multiply the running product by the next factor of 2.
    5. 5Continue multiplying by 2 two more times to finish all five factors.

    Common mistake. A common mistake is multiplying the base and exponent and answering 10. The exponent 5 tells you to use 2 as a factor five times, so 252^{5} means 2×2×2×2×22 \times 2 \times 2 \times 2 \times 2, not 2×52 \times 5.

    The idea underneath. A power is repeated multiplication: the exponent tells how many copies of the base to multiply.