Missing Bases, Missing Exponents, and Common Mistakes | Grade 6 Math | LevelKite

A power is a short way to write repeated multiplication. The base is the number that repeats in the multiplication. The exponent is the small raised number. It tells how many times the base is used as a factor, which is a number being multiplied. The result is the value of the power.

In this unit every exponent is a whole number, and in this lesson it is always 1 or more. The base may be a whole number, a fraction or a decimal. Here it is always positive, which means greater than 0.

You can already evaluate a power like the one below. Its base is $2$ , its exponent is $5$ and its value is $32$ . Today you will run that thinking backward to find a missing exponent or a missing base. Then you will use it to find mistakes in work with powers.

2^5 = \underbrace{2 \times 2 \times 2 \times 2 \times 2}_{5 \text{ factors of } 2} = 32

Key concept

The base repeats and the exponent counts

In a power with a whole-number exponent of 1 or more, the base is the factor that repeats. The exponent counts how many times the base appears. This is true whether the base is a whole number, a fraction or a decimal.

A power links three numbers: the base, the exponent and the value. Writing the power as repeated multiplication does not change its value, because that is what the power means. So when the exponent or the base is missing, write out the multiplication and ask what makes it true. When you check work, look for the right factor repeated the right number of times.

\text{base}^{\text{exponent}} = \text{value}

Finding a missing exponent. Look at $2^{\square} = 32$ . The box $\square$ marks the missing number. It sits in the exponent's place, so it stands for a whole number that counts factors. The question is: how many factors of $2$ multiply to make $32$ ?

To answer, start with one factor of the base. Keep multiplying by the base, and count the factors as you go. Stop when the product (the answer to the multiplication) equals the value. The count is the missing exponent. The steps are the same when the base is a fraction or a decimal.

In these problems the base is positive and is not $1$ . So each new factor makes the product bigger (base more than $1$ ) or smaller (base less than $1$ ). The products never repeat, so only one count matches the value. In this lesson the product always lands exactly on the value.

The table below counts the factors for $2^{\square} = 32$ . The product reaches $32$ at $5$ factors, so the missing exponent is $5$ .

Counting factors of 2 until the product is 32

Number of factors	Factors written out	Product
$1$	$2$	$2$
$2$	$2 \times 2$	$4$
$3$	$2 \times 2 \times 2$	$8$
$4$	$2 \times 2 \times 2 \times 2$	$16$
$5$	$2 \times 2 \times 2 \times 2 \times 2$	$32$

Worked example

Find a missing exponent with a fraction base

Find the missing exponent: $\left(\frac{1}{2}\right)^{\square} = \frac{1}{16}$ .

The box is in the exponent's place, so it stands for a whole number that counts factors. The parentheses show that the entire fraction 1/2 is the base. Ask: how many factors of 1/2 multiply to make 1/16?
$\left(\frac{1}{2}\right)^{\square} = \frac{1}{16}$
Start with one factor of the base. A single factor already counts as 1, so the count starts at 1.
$\text{1 factor: } \frac{1}{2}$
Multiply by the base again, so there are 2 factors. To multiply fractions, multiply the numerators together and the denominators together. The product 1/4 is not 1/16 yet, so keep going.
$\text{2 factors: } \frac{1}{2} \times \frac{1}{2} = \frac{1 \times 1}{2 \times 2} = \frac{1}{4}$
Multiply by the base again, so there are 3 factors. The product 1/8 is not 1/16 yet, so keep going.
$\text{3 factors: } \frac{1}{4} \times \frac{1}{2} = \frac{1 \times 1}{4 \times 2} = \frac{1}{8}$
Multiply by the base once more, so there are 4 factors. The product is exactly 1/16, so stop. The count of factors, 4, is the missing exponent.
$\text{4 factors: } \frac{1}{8} \times \frac{1}{2} = \frac{1 \times 1}{8 \times 2} = \frac{1}{16}$

Answer

The missing exponent is $4$ , because $\frac{1}{2}$ used as a factor $4$ times gives $\frac{1}{16}$ .

Finding a missing base. Look at $\square^2 = 49$ . Now the box is in the base's place. The whole-number exponent $2$ gives the count: the missing base is used as a factor $2$ times. The question is: which positive number, multiplied by itself, makes $49$ ?

To answer, test a base. Write out the factors and multiply. Test $6$ : $6 \times 6 = 36$ , which is too small. For positive bases with the same whole-number exponent of 1 or more, a bigger base gives a bigger value. So test a bigger base. Test $7$ : $7 \times 7 = 49$ . The missing base is $7$ .

The base does not have to be a whole number. When the value is a fraction or a decimal, test a fraction or a decimal as the base.

Worked example

Find a missing base that is a fraction

Find the missing base: $\square^3 = \frac{8}{27}$ .

The box is in the base's place. The whole-number exponent 3 counts factors, so the missing base is used as a factor 3 times. Ask: which positive number, used as a factor 3 times, makes 8/27?
$\square \times \square \times \square = \frac{8}{27}$
The value is a fraction, so test a fraction. As in the last example, fractions multiply numerator by numerator and denominator by denominator. So look for a numerator that gives 8 when it is used as a factor 3 times. The number 2 works.
$2 \times 2 \times 2 = 4 \times 2 = 8$
For the same reason, look for a denominator that gives 27 when it is used as a factor 3 times. The number 3 works.
$3 \times 3 \times 3 = 9 \times 3 = 27$
A correct base must make the equation true, so test 2/3 by putting it in the box. Write out all 3 factors and multiply. The product equals the value, so the equation is true.
$\frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} = \frac{2 \times 2 \times 2}{3 \times 3 \times 3} = \frac{8}{27}$

Answer

The missing base is $\frac{2}{3}$ , because $\frac{2}{3}$ used as a factor $3$ times gives $\frac{8}{27}$ .

Finding mistakes. The same meaning lets you check work that someone else wrote. Write the power as repeated multiplication yourself, then compare it with the work. Two questions guide you. They fit any power with a whole-number exponent of 1 or more, whether the base is a whole number, a fraction or a decimal.

In the work, is every factor the entire base?
In the work, is the number of factors equal to the exponent?

If either answer is no, you have found a mistake. If both answers are yes, check the multiplying too. The table shows three mistakes that the two questions catch.

Three mistakes in powers with whole-number exponents

Power	Mistaken work	What went wrong	Correct work
$2^5$	$5 \times 5 = 25$	The base and the exponent are swapped. The base $2$ should repeat, and there should be $5$ factors.	$2 \times 2 \times 2 \times 2 \times 2 = 32$
$5^3$	$5 \times 5 \times 5 \times 5 = 625$	There are $4$ factors. The exponent $3$ counts factors, not $\times$ signs.	$5 \times 5 \times 5 = 125$
$\left(\frac{2}{3}\right)^2$	$\frac{2 \times 2}{3} = \frac{4}{3}$	Only the numerator repeats. The entire fraction $\frac{2}{3}$ is the base.	$\frac{2}{3} \times \frac{2}{3} = \frac{4}{9}$

Worked example

Find and fix a mistake

Leo solved $\square^2 = 0.36$ . He wrote $0.36 \div 2 = 0.18$ and said the missing base is $0.18$ . Find Leo's mistake. Then find the correct base.

A correct answer must make the equation true, so test Leo's answer by putting 0.18 in the box. The whole-number exponent 2 means the base is used as a factor 2 times. Since 18 times 18 is 324 and the two factors have four decimal places in all, the product is 0.0324.
$0.18^2 = 0.18 \times 0.18 = 0.0324$
Compare the product with the value. They are not equal, so 0.18 is not the missing base.
$0.0324 \ne 0.36$
Name the mistake. Leo divided 0.36 by the exponent 2. Dividing by 2 undoes multiplying by 2, so it solves a different equation: the box times 2 equals 0.36. That uses 2 as a factor. In Leo's problem the 2 is a whole-number exponent. It counts the two factors, and both factors are the base.
$\square \times \square = 0.36 \quad \text{not} \quad \square \times 2 = 0.36$
Ask the right question: which positive number, multiplied by itself, makes 0.36? Since 6 times 6 is 36, test 0.6. Tenths times tenths are hundredths, so 0.6 times 0.6 is 36 hundredths, which is 0.36. The equation is true.
$0.6^2 = 0.6 \times 0.6 = 0.36$

Answer

Leo treated the exponent $2$ as a factor and divided by it. The correct missing base is $0.6$ , because $0.6 \times 0.6 = 0.36$ .

Takeaway. One idea did all the work today. In a power with a whole-number exponent of 1 or more, the base is the factor that repeats and the exponent counts the factors. This holds whether the base is a whole number, a fraction or a decimal.

To find a missing exponent, count the factors of the base that make the value.
To find a missing base, look for the positive number that makes the value when it is used as a factor as many times as the exponent says.
To find a mistake, write out the factors. Check that every factor is the entire base and that the number of factors equals the exponent.