Working with exponents and radicals

Exponents and radicals are two sides of one idea: an exponent repeats a base by multiplication, and a radical undoes it. A handful of rules cover almost everything the SAT asks. Multiplying powers with the same base adds the exponents, dividing subtracts them, and raising a power to a power multiplies them. Those rules apply across a product, never across a sum. A negative exponent is a reciprocal, and a fractional exponent is a root. The work is short, so the points leak in four predictable places: the wrong rule for combining powers, a power or root spread across a sum, a negative or fractional exponent misread, and the right number reported for the wrong quantity.

Three rules handle most exponent work, and all three are about what happens to the exponents while the base stays the same. Multiplying like bases adds the exponents. Dividing like bases subtracts them. Raising a power to a power multiplies them. The single most common slip is reaching for the wrong one of the three, usually multiplying the exponents when the rule called for adding.

• Same base, multiplied: add the exponents. $x^{3}\cdot x^{4}=x^{3+4}=x^{7}$.
• Same base, divided: subtract the exponents. $\dfrac{x^{9}}{x^{2}}=x^{9-2}=x^{7}$.
• Power of a power: multiply the exponents. $(x^{4})^{3}=x^{4\cdot 3}=x^{12}$.

Worked example. Simplify $\dfrac{(x^{3})^{2}\cdot x^{4}}{x^{5}}$.

This is the single biggest content trap on the topic. A power distributes across a product, so $(2x)^{3}=2^{3}x^{3}=8x^{3}$. It does not distribute across a sum: $(x+3)^{2}$ is not $x^{2}+9$. Squaring a sum means multiplying it out, $(x+3)(x+3)=x^{2}+6x+9$, and that middle term is exactly what the shortcut throws away. The same goes for roots: $\sqrt{x^{2}+9}$ is not $x+3$, because a root does not split across a sum either.

• Over a product, a power applies to each factor: $(ab)^{2}=a^{2}b^{2}$.
• Over a sum, you must multiply it out: $(a+b)^{2}=a^{2}+2ab+b^{2}$.
• A root behaves the same way: $\sqrt{a+b}$ is not $\sqrt{a}+\sqrt{b}$.

Worked example. Expand $(x+3)^{2}$.

Two kinds of exponent get misread because the symbol looks like it means something it does not. A negative exponent is not a negative number; it means a reciprocal. $x^{-3}=\dfrac{1}{x^{3}}$, and the value stays positive when $x$ is positive. A fractional exponent is a root: the denominator is the root and the numerator is the power. So $9^{\tfrac{1}{2}}=\sqrt{9}=3$, and $8^{\tfrac{2}{3}}=(\sqrt[3]{8})^{2}=2^{2}=4$. A half power is a square root, not half the base.

• Negative exponent: flip the base to the other side. $x^{-n}=\dfrac{1}{x^{n}}$, and $\dfrac{1}{x^{-n}}=x^{n}$.
• Unit fraction exponent: take the root named by the denominator. $a^{\tfrac{1}{n}}=\sqrt[n]{a}$.
• General fraction exponent: root by the denominator, power by the numerator. $a^{\tfrac{m}{n}}=(\sqrt[n]{a})^{m}$.

Worked example. Evaluate $8^{\tfrac{2}{3}}$.

The algebra on this topic is usually quick, so the trap moves to the last step. An equation with a power often gives you one quantity when the question asks for another. If $x^{2}=49$ is solved to $x=7$, an item might want $x$, or $x^{2}$, or $x+1$, or the value of an expression built on $x$. Reading off the wrong one produces a number that is real and even appears among the choices, but answers a different question. When a square is involved, watch whether the item wants the base or the squared value, and whether it restricts to the positive root.

Worked example. If $x^{2}=36$, what is $x^{2}+1$?

Pick the right rule for combining powers, keep a power or root from splitting across a sum, read a negative exponent as a reciprocal and a fractional one as a root, then report the exact quantity the item names.

Four patterns recur on exponents and radicals. They are the same ones the diagnostic routes on.