Grade 10 Math Exponents Lesson: Equivalent Algebraic Expressions and Properties of Exponents

Instructions

1. Review the six exponent properties in the Rainbow 6 reference.
2. Follow the lesson sequence: I Do, You Do, and We Do.
3. Show each step when generating an equivalent expression.
4. Complete the exit ticket independently.
5. Finish the 10-question practice worksheet. Use the optional challenge if time allows.

Learning goal: I can generate equivalent algebraic expressions using one or more properties of exponents.

Time plan: I Do: 4 minutes | You Do: 4 minutes | We Do: 4 minutes | Exit Ticket: 3 minutes

Rainbow 6: The Six Exponent Properties

Use the six colors as six stops in the game. At each stop, identify the property and rewrite the expression.

1. Red — Product of Powers

   • Same base multiplied: a^m · a^n = a^(m+n)
   • Example: x^3 · x^4 = x^7

2. Orange — Quotient of Powers

   • Same base divided: a^m ÷ a^n = a^(m−n)
   • Example: y^8 ÷ y^3 = y^5

3. Yellow — Power of a Power

   • A power raised to another power: (a^m)^n = a^(m·n)
   • Example: (p^2)^5 = p^10

4. Green — Power of a Product

   • A product raised to a power: (ab)^n = a^n b^n
   • Example: (3x)^2 = 3^2x^2 = 9x^2

5. Blue — Power of a Quotient

   • A quotient raised to a power: (a/b)^n = a^n/b^n
   • Example: (m/4)^3 = m^3/4^3 = m^3/64

6. Purple — Zero Exponent

   • Any nonzero base raised to zero equals 1: a^0 = 1
   • Example: 7^0 = 1

Helpful reminder: When multiplying like bases, add exponents. When raising a power to a power, multiply exponents.

I Do: Teacher Model

Watch the example and identify the Rainbow 6 property being used.

Example: Rewrite (x^3)^2 · x^4 as one power of x.

1. Use the power-of-a-power property: (x^3)^2 = x^6.
2. Multiply like bases: x^6 · x^4 = x^10.
3. Equivalent expression: x^10.

Property path: Yellow, then Red

Try this guided example. Fill in the missing steps.

(a^2)^3 · a^5

1. Power of a power: (a^2)^3 = a^___.
2. Product of powers: a^___ · a^5 = a^___.
3. Equivalent expression: a^___.

You Do: Solo Rainbow Stop

Rewrite each expression. Record the property used.

1. b^4 · b^3 = b^___ Property: ____

2. (c^2)^4 = c^___ Property: ____

3. (5d)^2 = __________ Property: ____

4. m^9 ÷ m^6 = m^___ Property: ____

We Do: Rainbow 6 Team Quest

Work with a partner or small group. For each expression, name the property or properties and generate an equivalent expression.

1. (q^2)^3 · q^4

   Property path: __

   Equivalent expression: __

2. (2r)^3 · r^2

   Property path: __

   Equivalent expression: __

3. (s^8 ÷ s^3)^2

   Property path: __

   Equivalent expression: __

4. Explain: Why is x^3 · x^4 equal to x^7, but x^3 + x^4 is not equal to x^7?

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Exit Ticket

Complete independently. Show your steps.

1. Name the property used in (z^4)^3 = z^12.

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2. Generate an equivalent expression for (3x^2)^2.

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3. Generate an equivalent expression for (a^3 · a^2) ÷ a^4.

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10-Question Practice Worksheet

Directions: Rewrite each expression in an equivalent form. Simplify when possible. For Questions 1–8, identify the exponent property or properties used.

1. x^5 · x^2

   Equivalent expression: ____ Property: ____

2. a^9 ÷ a^4

   Equivalent expression: ____ Property: ____

3. (m^3)^4

   Equivalent expression: ____ Property: ____

4. (4y)^2

   Equivalent expression: ____ Property: ____

5. (p/q)^3

   Equivalent expression: ____ Property: ____

6. 7^0

   Equivalent expression: ____ Property: ____

7. (b^2 · b^5) ÷ b^3

   Equivalent expression: ____ Property or properties: ____

8. (c^2d^3)^2

   Equivalent expression: ____ Property: ____

9. Real-world application: A video game awards 2^4 points at the start of a challenge. The points double during each of 3 rounds. Write an expression using exponents and simplify it.

   Expression: _____

   Simplified value: ___

10. Multi-property challenge: Simplify the expression. Show at least two steps.

    (x^3y^2 · x^4y^5) ÷ x^2y^3

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    Equivalent expression: ___

Optional Challenge

A student says that (2x^3)^2 = 4x^5. Is the student correct? Explain the error and write the correct equivalent expression.

Answer Key

I Do guided example

1. (a^2)^3 = a^6.
2. a^6 · a^5 = a^11.
3. Equivalent expression: a^11.

You Do

1. b^7; Product of Powers.
2. c^8; Power of a Power.
3. 25d^2; Power of a Product.
4. m^3; Quotient of Powers.

We Do

1. (q^2)^3 · q^4 = q^6 · q^4 = q^10.

   • Properties: Power of a Power, then Product of Powers.

2. (2r)^3 · r^2 = 2^3r^3 · r^2 = 8r^5.

   • Properties: Power of a Product, then Product of Powers.

3. (s^8 ÷ s^3)^2 = (s^5)^2 = s^10.

   • Properties: Quotient of Powers, then Power of a Power.

4. x^3 · x^4 represents multiplication, so the exponents add: x^7. The expression x^3 + x^4 represents addition, and exponent properties do not combine terms across addition. Therefore, x^3 + x^4 is not x^7.

Exit Ticket

1. Power of a Power.
2. (3x^2)^2 = 3^2(x^2)^2 = 9x^4.
3. (a^3 · a^2) ÷ a^4 = a^5 ÷ a^4 = a.

10-Question Practice Worksheet

1. x^7; Product of Powers.
2. a^5; Quotient of Powers.
3. m^12; Power of a Power.
4. 16y^2; Power of a Product.
5. p^3/q^3; Power of a Quotient.
6. 1; Zero Exponent.
7. (b^2 · b^5) ÷ b^3 = b^7 ÷ b^3 = b^4; Product of Powers and Quotient of Powers.
8. (c^2d^3)^2 = c^4d^6; Power of a Product and Power of a Power.
9. 2^4 · 2^3 = 2^7 = 128 points; Product of Powers.
10. (x^3y^2 · x^4y^5) ÷ x^2y^3 = (x^7y^7) ÷ x^2y^3 = x^5y^4; Product of Powers and Quotient of Powers.

Optional Challenge

The student is incorrect. The exponent on x must be multiplied by 2, not added to the outside coefficient. (2x^3)^2 = 2^2(x^3)^2 = 4x^6.