Математика20 мин

Exponents

Index laws, fractional and negative powers, and how to manipulate expressions like $a^{m/n}$ confidently.

The index laws

Every exponent problem reduces to four core laws. Memorise them once and they apply to every base.

a^m \cdot a^n = a^{m+n}

\dfrac{a^m}{a^n} = a^{m-n}

(a^m)^n = a^{mn}

(ab)^n = a^n b^n,\quad \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}

Laws only combine **same** bases. $2^3 \cdot 3^2$ does not simplify by adding exponents.

a^0 = 1\quad (a \neq 0)

a^{-n} = \dfrac{1}{a^n}

a^{1/n} = \sqrt[n]{a}

a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m

Evaluate

Compute $27^{2/3}$ .

Жауап: 9

Strategy: rewrite both sides with the same base, then equate exponents. If bases don't match, take logarithms.

Match the base

Solve $4^{x+1} = 32$ .

Жауап: $x = 3/2$

Жаттығу

Simplify $\dfrac{x^5 \cdot x^{-2}}{x^4}$ .

Жауап пен түсініктемені көру

Жауап: $x^{-1} = 1/x$

Numerator: $x^{5-2} = x^3$ . Then $x^3/x^4 = x^{-1}$ .