8th Grade Properties of Integer Exponents Lesson (Free PDF)

Q: Why does any nonzero number to the zero power equal 1?

Use the quotient rule: \frac{5^{3}}{5^{3}} = 1 and also 5^{3-3} = 5^{0}. Both must be equal, so 5^{0} = 1.

Q: Does a negative exponent make the answer negative?

No. It makes the power a reciprocal. For example 4^{-2} = \frac{1}{4^{2}} = \frac{1}{16}, which is positive.

Q: Can I add exponents when the bases are different?

No. The product rule only works for the same base. 2^{3} \cdot 3^{2} = 8 \cdot 9 = 72 and cannot be written as one power.

Q: What is the difference between (-3)^2 and -3^2?

\left(-3\right)^{2} = 9 because the base is -3. -3^{2} = -9 because only 3 is squared and then the result is negated.

Q: How do I simplify a power of a fraction with a negative exponent?

Flip the fraction and make the exponent positive: \left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^{2} = \frac{9}{4}.

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

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

An integer exponent tells how many times a base is multiplied by itself, or divided if the exponent is negative. To multiply powers with the same base, add the exponents; to divide, subtract them; for a power of a power, multiply them. Also, a nonzero base to the zero power equals 1.

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

Key facts in this integer exponents lesson

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: \(2^{-3} = \frac{1}{8}\).

What is an exponent?

An exponent tells how many times a base is used as a factor. In \(2^{5}\), the base is 2 and the exponent is 5, so \(2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32\). Integer exponents can also be zero or negative.

The five exponent properties

For nonzero \(a\) and integers \(m\) and \(n\): Product \(a^{m} \cdot a^{n} = a^{m+n}\). Quotient \(\frac{a^{m}}{a^{n}} = a^{m-n}\). Power \(\left(a^{m}\right)^{n} = a^{mn}\). Zero \(a^{0} = 1\). Negative \(a^{-n} = \frac{1}{a^{n}}\).

Why does the product rule add exponents?

Count the factors. \(3^{4}\) is four 3s and \(3^{2}\) is two more 3s, so the product has six 3s.

3^4 × 3^2 = 3^6

Why is a zero or negative exponent defined that way?

Look at the pattern in powers of 2. Each step to the right divides by 2, so the exponent drops by 1. Keep the pattern going and you reach \(2^{0} = 1\), then \(2^{-1} = \frac{1}{2}\), \(2^{-2} = \frac{1}{4}\), and so on.

Each step to the right divides by 2, so the exponent drops by 1.

How do you simplify an expression with exponents?

Work through the four steps in order.

Group the powers that have the same base.
Apply the product, quotient, or power rule.
Rewrite negative exponents as fractions.
Evaluate if the question asks for a number.

Worked example. Simplify \(\frac{5^{3} \cdot 5^{-5}}{5^{-2}}\). The numerator is \(5^{3 + (-5)} = 5^{-2}\). Then \(\frac{5^{-2}}{5^{-2}} = 5^{-2 – (-2)} = 5^{0} = 1\).

Square with side 3² in.

Area is a good check for the power rule: a square with side \(3^{2} = 9\) inches has area \(9^{2} = 81 = 3^{4}\) square inches, so \(\left(3^{2}\right)^{2} = 3^{4}\). Watch the signs: \(\left(-2\right)^{2} = 4\) but \(-2^{2} = -4\).

Powers of 10 are everywhere. A meter has \(10^{2} = 100\) centimeters, and a millimeter is \(10^{-3}\) meter. When you multiply \(10^{4} \cdot 10^{-6}\), you add the exponents to get \(10^{-2}\), which is 0.01. Practice the rules until each one feels automatic, and always check the sign of the exponent first.

Integer exponents lesson: questions and answers

Why does any nonzero number to the zero power equal 1?

Use the quotient rule: \(\frac{5^{3}}{5^{3}} = 1\) and also \(5^{3-3} = 5^{0}\). Both must be equal, so \(5^{0} = 1\).

Does a negative exponent make the answer negative?

No. It makes the power a reciprocal. For example \(4^{-2} = \frac{1}{4^{2}} = \frac{1}{16}\), which is positive.

Can I add exponents when the bases are different?

No. The product rule only works for the same base. \(2^{3} \cdot 3^{2} = 8 \cdot 9 = 72\) and cannot be written as one power.

What is the difference between (-3)^2 and -3^2?

\(\left(-3\right)^{2} = 9\) because the base is \(-3\). \(-3^{2} = -9\) because only 3 is squared and then the result is negated.

How do I simplify a power of a fraction with a negative exponent?

Flip the fraction and make the exponent positive: \(\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^{2} = \frac{9}{4}\).

How to use this integer exponents lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

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