Apply exponent laws to simplify expressions

Exponent laws describe how multiplication, division, and powers interact: $a^m a^n=a^{m+n}$ , $\frac{a^m}{a^n}=a^{m-n}$ , and $(a^m)^n=a^{mn}$ , including zero, negative, and rational exponents where defined. Simplification requires tracking nonzero-base and domain restrictions, recognizing that exponents do not distribute over sums or differences; these laws provide the algebraic foundation for working with exponential models, logarithms, and equations.

Detailed Explanation: 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.

Learn by doing: Apply exponent laws to simplify expressions

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Algebra with Exponents - Monomial and Constant

Earned ?