Expressions

Simplifying with Exponents

Simplify a product of powers with the same base by applying the product rule for exponents.

Example

Use product and power rules to simplify powers of x.

highlighted = computed this step

Step 1 — Product rule setup

Set up the expression.

x3×x4x^{3} \times x^{4}
Product rule: the base is the same x.same base: add exponentsx^3*x^4same base x--power of a power: multiply exponents(x^2)^3inner 2outer 3multiply exponents--Product example: x^3 * x^4 -> x^7Power example: (x^2)^3 -> x^6

Step 2 — Add exponents

Same base product: add exponents, 3 + 4 = 7.

3+4=73 + 4 = \hl{7}
Add exponents: 3 + 4 = 7.same base: add exponentsx^3*x^43 + 4 = 7--power of a power: multiply exponents(x^2)^3inner 2outer 3multiply exponents--Product example: x^3 * x^4 -> x^7Power example: (x^2)^3 -> x^6

Step 3 — Product result

Write the product as x^7.

x7\hlmath{x^{7}}
Product result: x^7.same base: add exponentsx^3*x^43 + 4 = 7x^7power of a power: multiply exponents(x^2)^3inner 2outer 3multiply exponents--Product example: x^3 * x^4 -> x^7Power example: (x^2)^3 -> x^6

Step 4 — Power setup

Set up the expression.

(x2)3( x^{2} )^{ 3 }
Power of a power: look at the inner and outer exponents.same base: add exponentsx^3*x^43 + 4 = 7x^7power of a power: multiply exponents(x^2)^3inner 2outer 3multiply exponents--Product example: x^3 * x^4 -> x^7Power example: (x^2)^3 -> x^6

Step 5 — Multiply exponents

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

2×3=62 \times 3 = \hl{6}
Multiply exponents: 2 x 3 = 6.same base: add exponentsx^3*x^43 + 4 = 7x^7power of a power: multiply exponents(x^2)^3inner 2outer 32 x 3 = 6--Product example: x^3 * x^4 -> x^7Power example: (x^2)^3 -> x^6

Step 6 — Power result

Write the power as x^6.

x6\hlmath{x^{6}}
Power result: x^6.same base: add exponentsx^3*x^43 + 4 = 7x^7power of a power: multiply exponents(x^2)^3inner 2outer 32 x 3 = 6x^6Product example: x^3 * x^4 -> x^7Power example: (x^2)^3 -> x^6
simplify-with-exponents Product rule: x^a · x^b = x^(a+b) — add exponents. Power-of-power rule: (x^a)^b = x^(ab) — multiply exponents.