Exponents and Roots

Introduction to Exponent Rules

Apply the product rule (b^m · b^n = b^(m+n)) and power rule ((b^m)^n = b^(m·n)).

exponent-rules-intro Key exponent rules for the same base b: - Product rule: b^m · b^n = b^(m+n). - Power rule: (b^m)^n = b^(m·n).

Example

Use exponent arithmetic for products and powers with the same base.

highlighted = computed this step

Step 1 — Product rule setup

Set up the expression.

23×222^{3} \times 2^{2}
2^3 times 2^2: write the factor groups in order.2223 factorsx222 factorsThe exponent on each part tells how many base factors it contributes.

Step 2 — Add exponents

Same base product: add exponents, 3 + 2 = 5.

3+2=53 + 2 = \hl{5}
Product rule counts all factors: 3 + 2 = 5.2223 factorsx222 factors3 + 2 = 5 factorsThe same base lets the factors line up into one stack of 5.

Step 3 — Evaluate product

Evaluate the power: 2^5 = 32.

25=322^{5} = \hl{32}
5 repeated factors give 2^5 = 32.2223 factorsx222 factors3 + 2 = 5 factors2 x 2 x 2 x 2 x 2 = 322^5 = 32Counting factors first gives the value 32.

Step 4 — Power rule setup

Set up the expression.

(23)2( 2^{3} )^{ 2 }
(2^3)^2: repeat the inside power 2 times.2223 factorscopy 12223 factorscopy 2The outside exponent makes 2 copies of the inside factor stack.

Step 5 — Multiply exponents

Power of a power: multiply exponents, 3 x 2 = 6.

3×2=63 \times 2 = \hl{6}
Power rule counts 2 groups of 3: 3 x 2 = 6.2223 factorscopy 12223 factorscopy 23 x 2 = 6 factorsThere are 2 stacks, each with 3 factors.

Step 6 — Evaluate power

Evaluate the power: 2^6 = 64.

26=642^{6} = \hl{64}
6 repeated factors give 2^6 = 64.2223 factorscopy 12223 factorscopy 23 x 2 = 6 factors2 x 2 x 2 x 2 x 2 x 2 = 642^6 = 64Counting factors first gives the value 64.

Step 7 — Result

Read the final result.

326432 \quad 64
Product result 32; power result 64.2223 factorsx222 factors3 + 2 = 5 factors2 x 2 x 2 x 2 x 2 = 322^5 = 32Both rules are just counting repeated factors.