Equivalent expressions

Equivalent expressions are algebraic expressions that simplify to the exact same value for any given input. SAT questions on this topic test your ability to manipulate expressions through expanding, factoring, combining like terms, and working with exponents or radicals.

Key Algebraic Expansion & Factoring Rules

• Distributive Property: $a(b + c) = ab + ac$

• Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$

• Perfect Square Trinomials:

  • $(a + b)^2 = a^2 + 2ab + b^2$

  • $(a - b)^2 = a^2 - 2ab + b^2$

• Factoring Quadratic Expressions: $x^2 + (p + q)x + pq = (x + p)(x + q)$

Exponent & Radical Rules

• Product Rule: $x^a \cdot x^b = x^{a+b}$

• Quotient Rule: $\dfrac{x^a}{x^b} = x^{a-b}$

• Power Rule: $(x^a)^b = x^{a \cdot b}$

• Negative Exponents: $x^{-a} = \dfrac{1}{x^a}$

• Rational Exponents & Radicals: $x^{a/b} = \sqrt[b]{x^a} = (\sqrt[b]{x})^a$

Rational Expressions

To add or subtract algebraic fractions, find a common denominator. To simplify, factor both the numerator and denominator to cancel common factors:

1. Match Coefficients After Expanding. When two expressions are stated to be equal for all values of $x$, expand both sides completely and match the corresponding coefficients of like terms. 

   Example: If $(2x + 3)(x - 4) = 2x^2 + ax + b$ for all values of $x$, what is the value of $a + b$?

   • Expand the left side using FOIL:

     $$(2x + 3)(x - 4) = 2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12$$

   • Match coefficients with $2x^2 + ax + b$:

     • $a = -5$

     • $b = -12$

   • Calculate $a + b$:

     $$a + b = -5 + (-12) = -17$$

2. Plug in a Test Value for $x$. If you get stuck manipulating an expression, substitute a small, easy number (like $x = 2$ or $x = 0$) into the given expression and into the answer choices. The equivalent expression will yield the exact same value. Avoid $x = 0$ or $x = 1$ if they make multiple choices evaluate to the same number. 

   Example: Which expression is equivalent to $\dfrac{2x^2 + 6x}{2x}$ for $x \neq 0$?

   • Test $x = 3$:

     $$\dfrac{2(3)^2 + 6(3)}{2(3)} = \dfrac{18 + 18}{6} = \dfrac{36}{6} = 6$$

   • Simplify algebraically to verify:

     $$\dfrac{2x(x + 3)}{2x} = x + 3$$

   • Evaluating $x + 3$ at $x = 3$ yields $3 + 3 = 6$.

3. Convert Rational Exponents to Radical Form. To simplify or match fractional exponents, rewrite them using radical notation ($x^{a/b} = \sqrt[b]{x^a}$). 

   Example: Which expression is equivalent to $(8x^6)^{2/3}$ for $x > 0$?

   • Apply the power rule to both the coefficient and the variable term:

     $$(8)^{2/3} \cdot (x^6)^{2/3}$$

   • Evaluate $8^{2/3}$:

     $$8^{2/3} = (8^{1/3})^2 = (2)^2 = 4$$

   • Multiply exponent powers for $x$:

     $$(x^6)^{2/3} = x^{6 \cdot \frac{2}{3}} = x^4$$

   • Combine the simplified terms:

     $$4x^4$$