Powers · Capstone · C1

Pick the right move.

The final skill is choosing among all the laws: add exponents to multiply, subtract to divide, multiply for a power of a power, raise every factor in a product, and turn negatives into reciprocals.

01 / THE IDEA

See it work.

Watch the method

Pick the right move

Read the expression, pick the law, simplify — then check for positive exponents.

1 / 1
(2x2)3(2x^{2})^{3}
1

Multiply the exponents

Power of a power: multiply the exponents.

(2x2)3=8x6(2x^{2})^{3} = 8 x^{6}
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Work it out.

Type with ^ for powers — e.g. x^5.

streak 0
best 0
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— / WHY IT WORKS

The idea underneath.

The whole toolkit

Multiply → add. Divide → subtract. Power of a power → multiply. Product → raise each factor. Zero → 1. Negative → reciprocal.

Why it works

One toolkit, the right tool each time.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Simplify y^4 · y^2

1

Add the exponents

Same base y — keep the base, add the exponents: 4 + 2. y^4 · y^2 = y^6

2

Write the product

Same base when multiplying → add exponents. y^4 · y^2 = y^6

EX 2medium

Simplify x^5 ÷ x^3

1

Subtract the exponents

Same base x — keep the base, subtract: 5 − 3. x^5 ÷ x^3 = x^2

2

Write the quotient

Same base when dividing → subtract exponents. \fracx^5x^3 = x^2

EX 3hard

Simplify x^0

1

Zero exponent

Any non-zero base to the power 0 is 1 (it comes from x^a ÷ x^a). x^0 = 1

04 / COMMON TRAPS

Where students slip.

Trap

Confusing add vs multiply exponents.

Fix

Multiplying powers adds; a power of a power multiplies.

Trap

Leaving negative exponents.

Fix

Rewrite as positive in the denominator.

05 / QUIZ

Test yourself.

The Test

12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/12Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).
Core

Core (4)

C1

Simplify y^0

Need a hint?
  • Identify the operation.
  • Apply its law; finish with positive exponents.
C2

Simplify (y^4)^4

Need a hint?
  • Identify the operation.
  • Apply its law; finish with positive exponents.
C3

Simplify z^6 ÷ z^2

Need a hint?
  • Identify the operation.
  • Apply its law; finish with positive exponents.
C4

Simplify y^5 · y^2

Need a hint?
  • Identify the operation.
  • Apply its law; finish with positive exponents.
Apply

Apply (4)

A1

Simplify z^2 · z^4

Need a hint?
  • Identify the operation.
  • Apply its law; finish with positive exponents.
A2

Simplify y^0

Need a hint?
  • Identify the operation.
  • Apply its law; finish with positive exponents.
A3

Simplify z^0

Need a hint?
  • Identify the operation.
  • Apply its law; finish with positive exponents.
A4

Simplify y^0

Need a hint?
  • Identify the operation.
  • Apply its law; finish with positive exponents.
Extension

Extension (4)

E1

Simplify x^5 ÷ x^1

Need a hint?
  • Identify the operation.
  • Apply its law; finish with positive exponents.
E2

Simplify (z^3)^2

Need a hint?
  • Identify the operation.
  • Apply its law; finish with positive exponents.
E3

Simplify y^3 · y^4

Need a hint?
  • Identify the operation.
  • Apply its law; finish with positive exponents.
E4

Simplify y^7 ÷ y^3

Need a hint?
  • Identify the operation.
  • Apply its law; finish with positive exponents.