8th Grade Properties of Integer Exponents Worksheet Solutions (PDF)

8th Grade math worksheet solutions: Properties of Integer Exponents – Maths PDF

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This integer exponents answers for 8th Grade covers properties of integer exponents (Common Core 8.EE.A.1), with a free PDF to print.

This answer key gives a full worked solution for each of the 21 problems on the integer exponents worksheet (Common Core 8.EE.A.1). Compare each step with your own work, not only the final answer, and note any step where your method differed.

Grade 8 · Common Core State Standards: 8.EE.A.1

Integer exponents answers: exercises 1 to 10

Exercise 1 – Product rule

Same base, add exponents: \(5^{3+4} = 5^{7}\).

\(5^{7} = 78{,}125\).

\(5^{7} = 78{,}125\)

Exercise 2 – Quotient rule

Subtract exponents: \(5^{7-4} = 5^{3}\).

\(5^{3} = 125\)

Exercise 3 – Power of a power

Multiply exponents: \(2^{3 \cdot 2} = 2^{6}\).

\(2^{6} = 64\)

Exercise 4 – Zero exponent

Any nonzero number to the zero power is 1.

1 and 1

Exercise 5 – Count the factors

2^3 × 2^2 = 2^5

The picture shows 3 + 2 = 5 factors of 2.

\(2^{3} \cdot 2^{2} = 2^{5}\).

\(2^{5} = 32\)

Exercise 6 – Negative exponent

Each step right divides by 3: \(3^{0} = 1\), then \(1 \div 3 = \frac{1}{3}\), then \(\frac{1}{3} \div 3 = \frac{1}{9}\).

Each step to the right divides by 3, so the exponent drops by 1.

\(3^{-1} = \frac{1}{3},\ 3^{-2} = \frac{1}{9}\)

Exercise 7 – Powers of ten

Each step to the right divides by 10, so the exponent drops by 1.

\(10^{-2} = \frac{1}{10^{2}} = \frac{1}{100}\).

\(\frac{1}{100} = 0.01\)

Exercise 8 – Mixed exponents

Add exponents: \(5 + (-3) = 2\).

\(2^{2} = 4\)

Exercise 9 – Negative power of a power

Multiply exponents: \(3^{-2} = \frac{1}{3^{2}}\).

\(\frac{1}{9}\)

Exercise 10 – Quotient with a negative result

Subtract exponents: \(4^{2-5} = 4^{-3}\).

\(4^{-3} = \frac{1}{4^{3}} = \frac{1}{64}\).

\(\frac{1}{64}\)

Exercises 11 and up

Exercise 11 – Power of a product

Way 1: \(10^{3} = 1{,}000\).

Way 2: \(2^{3} \cdot 5^{3} = 8 \cdot 125 = 1{,}000\).

Both ways give 1,000.

Exercise 12 – Fraction to a negative power

Flip the fraction: \(\left(\frac{3}{2}\right)^{3} = \frac{27}{8}\).

\(\frac{27}{8}\)

Exercise 13 – Several rules

Numerator: \(5^{-3+6} = 5^{3}\).

Then \(\frac{5^{3}}{5^{2}} = 5^{1}\).

\(5\)

Exercise 14 – Write as a power

\(81 = 3^{4}\), so \(\frac{1}{81} = \frac{1}{3^{4}} = 3^{-4}\).

\(3^{-4}\)

Exercise 15 – Power of a power with negatives

\(\left(2^{-3}\right)^{2} = 2^{-6}\). Then \(2^{-6} \cdot 2^{9} = 2^{3}\).

\(2^{3} = 8\)

Exercise 16 – Nested rules

Numerator: \(4^{2 + (-5)} = 4^{-3}\).

Divide: \(4^{-3 – (-1)} = 4^{-2} = \frac{1}{16}\).

\(\frac{1}{16}\)

Exercise 17 – Find the exponent

Add exponents: \(n + 5 = 2\), so \(n = -3\).

Check: \(3^{-3} \cdot 3^{5} = 3^{2} = 9\).

\(n = -3\)

Exercise 18 – Bacteria growth

Each bar is twice the one before.

After 10 hours: \(2^{10} = 1{,}024\) cells.

Compare: \(\frac{2^{12}}{2^{10}} = 2^{2} = 4\).

1,024 cells; 4 times as many

Exercise 19 – Computer storage

\(\frac{2^{30}}{2^{10}} = 2^{20}\).

\(2^{20} = 1{,}048{,}576\).

\(2^{20} = 1{,}048{,}576\) kilobytes

Exercise 20 – Square tile

Area \(= \left(3^{2}\right)^{2} = 3^{4} = 81\).

\(3^{4} = 81\) square inches

Exercise 21 – Find the error

She added the exponents. For a power of a power, multiply them: \(5^{2 \cdot 3} = 5^{6}\).

\(5^{6} = 15{,}625\).

\(5^{6} = 15{,}625\)

Key facts behind this integer exponents answers

Key facts

Product rule: \(a^{m} \cdot a^{n} = a^{m+n}\).
Quotient rule: \(\frac{a^{m}}{a^{n}} = a^{m-n}\).
Power rule: \(\left(a^{m}\right)^{n} = a^{mn}\).
\(a^{0} = 1\) and \(a^{-n} = \frac{1}{a^{n}}\) for nonzero \(a\).
A negative exponent does not make the number negative: \(2^{-3} = \frac{1}{8}\).

How to use this integer exponents answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

Download the integer exponents answers as a PDF with the button below and keep it with your notes.

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