Product and Power Rules for Exponents

The Product Rule says x^a * x^b = x^(a+b): add exponents when multiplying the same base. The Power Rule says (x^a)^b = x^(a*b): multiply exponents when raising a power to a power. The Product-to-Power Rule says (xy)^n = x^n y^n.

Example

Use exponent arithmetic for product, power, and product-to-power rules.

highlighted = computed this step

Step 1 — Product rule

Set up the expression.

x^{3} \times x^{4}

Step 2 — Add exponents

Add exponents: 3 + 4 = 7.

3 + 4 = \hl{7}

Step 3 — Product result

Read the final result.

\hlmath{x^{7}}

Step 4 — Power rule

Set up the expression.

( x^{2} )^{ 3 }

Step 5 — Multiply exponents

Multiply exponents: 2 x 3 = 6.

2 \times 3 = \hl{6}

Step 6 — Power result

Read the final result.

\hlmath{x^{6}}

Step 7 — Product to power

Apply the exponent 2 to both factors: 2 and 2.

(xy)^ { 2 }=x^{ \hl{2} }y^{ \hl{2} }

product-power-rules Three exponent rules in one lesson: product, power, and product-to-power. Each rule manipulates exponents arithmetically. Apply the correct rule by recognizing the form of the expression.