Integrated Math 2–3 Honors · Lesson Review · September 1, 2026

Laws of Exponents and Simplifying Expressions

Build each exponent rule from repeated multiplication, then simplify products, quotients, powers, negative exponents, and rational monomials.

30-minute edited lesson11 chaptersCaptions + transcript

01 · Watch

Lesson video

VTT captions SRT captions

0:0001Laws of Exponents 0:0502Repeated Addition and Multiplication 2:1303Product Rule for Exponents 8:4004Distribute, Then Combine Bases 9:5805Quotient Rule for Exponents 13:0306Negative Exponents 14:5907Simplifying Rational Monomials 18:4808Powers of Products and Quotients 22:0909A Multi-Step Power Example 26:2010Negative-Exponent Practice 27:5411The Zero-Exponent Rule

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02 · Understand

Learning targets

Multiply powers with the same base by adding exponents.
Divide powers with the same base by subtracting exponents.
Apply an outside exponent to every factor in a product or quotient.
Rewrite final answers using positive exponents.

03 · Rules

Exponent laws

Read the structure first; then choose the matching rule.

01Product rule: For the same nonzero base, $x^a\cdot x^b=x^{a+b}$ .
02Quotient rule: For the same nonzero base, $\frac{x^a}{x^b}=x^{a-b}$ .
03Power of a power: Multiply the exponents: $(x^a)^b=x^{ab}$ .
04Power of a product: Apply the exponent to every factor: $(xy)^a=x^ay^a$ .
05Power of a quotient: Apply the exponent to numerator and denominator: $\left(\frac{x}{y}\right)^a=\frac{x^a}{y^a}$ .
06Negative exponent: Move the factor across the fraction bar: $x^{-a}=\frac1{x^a}$ .
07Zero exponent: Every nonzero base to the zero power is one: $x^0=1\quad(x\ne0)$ .

04 · Apply

Worked examples

Simplify coefficients and each variable factor separately.

Product rule

$x^4\cdot x^5$

x^4\cdot x^5=x^{4+5}=x^9

The base stays the same. Add the counts of repeated factors.

Rational monomial

$\frac{2x^3y^5}{8xy^8}$

Reduce the coefficients: $\frac28=\frac14$ .
Subtract exponents on matching bases: $x^{3-1}=x^2$ and $y^{5-8}=y^{-3}$ .
Write the answer with positive exponents: $\frac{x^2}{4y^3}$ .

Power of a quotient

$\left(\frac{3x^2y^3}{2z^4}\right)^3$

\left(\frac{3x^2y^3}{2z^4}\right)^3=\frac{27x^6y^9}{8z^{12}}

Cube every coefficient and variable factor; multiply the nested exponents.

Negative exponents

$\frac{3x^4y^2}{x^{-5}y^3}$

\frac{3x^4y^2}{x^{-5}y^3}=3x^{4-(-5)}y^{2-3}=\frac{3x^9}{y}

Subtracting a negative exponent adds; the remaining negative power moves to the denominator.

05 · Avoid

Common mistakes

Adding exponents on unlike bases.

The product rule requires the same base.

Adding instead of multiplying nested exponents.

In $(x^a)^b$ , use $ab$ .

Applying a power to only one factor.

An outside exponent reaches every factor inside parentheses.

Losing the subtraction sign.

For quotients, write the exponent difference before simplifying.

Leaving negative exponents.

Move the factor across the fraction bar for a positive exponent.

Using the zero rule at zero.

$x^0=1$ requires $x\ne0$ .

06 · Check yourself

Quick check

Name the rule before opening each answer.

1Simplify

m^7\cdot m^{-2}

$m^5$ because $7+(-2)=5$ .

2Simplify

\frac{p^3}{p^8}

with positive exponents.

$\frac1{p^5}$ because $p^{3-8}=p^{-5}$ .

3Simplify

(2a^3b)^2

$4a^6b^2$ . Square every factor.

4Why does

x^0=1

From the quotient rule, $\frac{x^a}{x^a}=x^{a-a}=x^0$ , while any nonzero quantity divided by itself equals one.

07 · Revisit

Transcript and captions

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