Exponents, Roots and Scientific Notation flashcards

No. An exponent counts how many times the base is used as a factor: $3^4=3\times 3\times 3\times 3=81$. It is repeated multiplication, not base times exponent.

Same base: multiply by adding exponents, $a^m\cdot a^n=a^{m+n}$; divide by subtracting, $\frac{a^m}{a^n}=a^{m-n}$ with a ≠ 0. Example: $x^6\cdot x^2=x^8$ and $\frac{x^6}{x^2}=x^4$.

$(a^m{)}^n=a^{mn}$: multiply the exponents, so $(y^3{)}^4=y^{12}$. $(ab{)}^n=a^n{b}^n$: the outside exponent applies to every factor, so $(2x{)}^3=8x^3$.

For nonzero a, $a^0=1$ and $a^{-n}=\frac{1}{a^n}$. So $7^0=1$ and $4^{-2}=\frac{1}{16}$. A negative exponent makes a reciprocal, never a negative number.

The radical gives the principal, nonnegative square root: $\sqrt{64}=8$, not −8; square roots of negatives are not real. Cube roots of negatives are real and negative: $\sqrt[3]{-27}=-3$.

Pull out the largest perfect-square factor using $\sqrt{ab}=\sqrt{a}\sqrt{b}$: $\sqrt{75}=\sqrt{25}\sqrt{3}=5\sqrt{3}$. Perfect squares to know: 4, 9, 16, 25, 36, 49, 64, 81, 100, ...

A number written as $c\times 10^n$ with $1\le c<10$. Large numbers get positive n and small numbers negative n: 48,000 = $4.8\times 10^4$ and 0.0062 = $6.2\times 10^{-3}$.

Multiply (divide) the front numbers and add (subtract) the exponents, then adjust so the front is at least 1 and under 10: $(4\times 10^5)(5\times 10^2)=20\times 10^7=2\times 10^8$.