Powers · Combine the laws · 7.3

Laws mastery.

Fluency means applying the right law instantly — add, subtract, or multiply exponents, handle a zero or negative exponent — across a mixed set.

01 / THE IDEA

See it work.

Watch the method

Laws mastery

Identify, apply, simplify — every time.

1 / 2
x7x4\frac{x^{7}}{x^{4}}
1

Subtract the exponents

Same base x — keep the base, subtract: 7 − 4.

x7÷x4=x3x^{7} \div x^{4} = x^{3}
2

Write the quotient

Same base when dividing → subtract exponents.

x7x4=x3\frac{x^{7}}{x^{4}} = x^{3}
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
… =
— / WHY IT WORKS

The idea underneath.

Automatic laws

With practice the rules become automatic: the operation tells you whether to add, subtract, or multiply the exponents.

Why it works

Operation → rule → simplify.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Simplify y^6 ÷ y^3

1

Subtract the exponents

Same base y — keep the base, subtract: 6 − 3. y^6 ÷ y^3 = y^3

2

Write the quotient

Same base when dividing → subtract exponents. \fracy^6y^3 = y^3

EX 2medium

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

EX 3hard

Simplify z^2 · z^2

1

Add the exponents

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

2

Write the product

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

04 / COMMON TRAPS

Where students slip.

Trap

Hesitating on which rule.

Fix

The operation decides the rule.

Trap

Leaving a negative exponent.

Fix

Finish with positive exponents.

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 (x^2)^3

Need a hint?
  • Read the operation.
  • Apply its rule and simplify.
C2

Simplify (y^2)^4

Need a hint?
  • Read the operation.
  • Apply its rule and simplify.
C3

Simplify x^3 ÷ x^1

Need a hint?
  • Read the operation.
  • Apply its rule and simplify.
C4

Simplify y^5 ÷ y^1

Need a hint?
  • Read the operation.
  • Apply its rule and simplify.
Apply

Apply (4)

A1

Simplify y^2 ÷ y^1

Need a hint?
  • Read the operation.
  • Apply its rule and simplify.
A2

Simplify y^6 ÷ y^2

Need a hint?
  • Read the operation.
  • Apply its rule and simplify.
A3

Simplify y^5 ÷ y^3

Need a hint?
  • Read the operation.
  • Apply its rule and simplify.
A4

Simplify z^0

Need a hint?
  • Read the operation.
  • Apply its rule and simplify.
Extension

Extension (4)

E1

Simplify x^5 · x^2

Need a hint?
  • Read the operation.
  • Apply its rule and simplify.
E2

Simplify y^0

Need a hint?
  • Read the operation.
  • Apply its rule and simplify.
E3

Simplify x^5 · x^2

Need a hint?
  • Read the operation.
  • Apply its rule and simplify.
E4

Simplify y^0

Need a hint?
  • Read the operation.
  • Apply its rule and simplify.