Skill: Apply exponent laws for products

When multiplying powers with the same base, add the exponents:

$$a^m\cdot a^n=a^{m+n}$$

When a product is raised to a power, apply the exponent to each factor:

$$(ab)^n=a^n b^n$$

Worked example

Simplify:

$$x^3\cdot x^2\cdot(2x)^2$$

Step 1: Multiply the powers with the same base.

The first two factors both have base $x$, so add their exponents:

$$x^3\cdot x^2=x^{3+2}=x^5$$

Now the expression is

$$x^5\cdot(2x)^2$$

Step 2: Apply the exponent to each factor inside the parentheses.

$$(2x)^2=2^2x^2=4x^2$$

So the expression becomes

$$x^5\cdot4x^2$$

Step 3: Multiply the powers with base $x$.

Add the exponents:

$$x^5\cdot4x^2=4x^{5+2}=4x^7$$

Therefore,

$$\boxed{4x^7}$$

Remember: when multiplying powers with the same base, add the exponents. When raising a product to a power, apply the exponent to every factor.