Skill: Apply exponent laws to simplify expressions

Exponent laws let you combine powers with the same nonzero base:

• Multiplication: $a^m a^n=a^{m+n}$
• Division: $\dfrac{a^m}{a^n}=a^{m-n}$
• Power of a power: $(a^m)^n=a^{mn}$
• Negative exponent: $a^{-n}=\dfrac{1}{a^n}$, where $a\ne0$

Exponents do not distribute over addition or subtraction. For example, $(a+b)^2\ne a^2+b^2$.

Worked example

Simplify

$$\frac{(x^3y^{-2})^2x^4}{x^{-1}y^3}.$$

Assume $x\ne0$ and $y\ne0$, since the expression contains negative exponents and division.

Step 1: Apply the power of a product and power of a power laws.

$$(x^3y^{-2})^2=x^{3\cdot2}y^{-2\cdot2}=x^6y^{-4}.$$

So the expression becomes

$$\frac{x^6y^{-4}x^4}{x^{-1}y^3}.$$

Step 2: Combine powers in the numerator.

Using $a^ma^n=a^{m+n}$,

$$x^6x^4=x^{10}.$$

Therefore,

$$\frac{x^{10}y^{-4}}{x^{-1}y^3}.$$

Step 3: Divide powers with the same base.

Subtract exponents:

$$\frac{x^{10}}{x^{-1}}=x^{10-(-1)}=x^{11},$$

and

$$\frac{y^{-4}}{y^3}=y^{-4-3}=y^{-7}.$$

Thus,

$$x^{11}y^{-7}.$$

Step 4: Rewrite the negative exponent.

Since $y^{-7}=\dfrac{1}{y^7}$,

$$\boxed{\frac{x^{11}}{y^7}}.$$

When simplifying, apply one exponent law at a time, carefully track signs, and rewrite negative exponents in the denominator if a positive-exponent answer is required.