Apply Properties of Integer Exponents

Students extend exponent rules—product rule, quotient rule, power of a power, and zero exponent—to negative integer exponents. They simplify expressions such as 3⁻² × 3⁵ and (2³)⁻¹ by applying these properties systematically, writing answers with positive exponents.

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Grade 8 · Expressions and Equations: Exponents and Scientific Notation · question 1 of 3

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20 rungs. A star at the top.

5^1 × 5^6 = 5

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A negative exponent writes a reciprocal, not a negative value.

A zero exponent on a nonzero base is 1.

Read it: Exponent rules by counting factors

2^2 × 2^3 is 2 factors of 2 and 3 more: (2 × 2) × (2 × 2 × 2), 5 factors in all. So 2^2 × 2^3 = 2^5: the exponents add.

For a product of one base, add the exponents.
For a quotient of one base, take the bottom exponent from the top one.
For a power of a power, multiply the exponents.

An exponent counts factors: 2^3 is 2 × 2 × 2, three factors of 2.

Multiplying joins the factors, so their counts add: 2 + 3 = 5.

Any nonzero number to the 0 is 1: 2^3 ÷ 2^3 cancels every factor and leaves 1.

Same base: multiplying adds the exponents, dividing takes them away, and a power of a power multiplies them.

Multiplying the exponents of a product: 2^2 × 2^3 = 2^6.

Count the factors: 2 + 3 = 5, so 2^5.

5^5 × 5^4 = 5^___ — 9. 5 factors and 4 more make 9: the exponents add, 5 + 4 = 9.

3⁻² × 3⁵

When you multiply two powers that share the same base, you don't have to expand them — there's a much faster way.

Let's simplify three to the negative two, times three to the fifth.

Here's the product rule: when bases match, keep the base and add the exponents.

Look — both powers have base three. So the rule applies, even though one exponent is negative.

Keep the three. Then add the exponents: negative two plus five.

Negative two plus five is three.

So the expression simplifies to three cubed.

Three cubed means three times three times three — which is twenty-seven.

Grade 8 · Same base? Add the exponents.

Three to the negative two times three to the fifth equals twenty-seven. The same-base shortcut works for negative exponents too!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics

Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal: 1 star — one skill mastered, every rung of its ladder climbed
Climbing now: Divide within 100

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MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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