An index is just counting.

$a^3$ means three $a$'s multiplied together — nothing more. Keep that picture of tiles in your head and every law of indices becomes obvious.

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$a^3 \times a^2$ is three tiles next to two: five tiles, $a^5$. Multiplying powers adds the indices, because all you're doing is counting tiles.

Dividing cancels tiles in pairs. $a^5 \div a^2$ leaves three on top. $a^2 \div a^5$ leaves three underneath: that's what $a^{-3}$ means. Cancel everything and you get $a^0 = 1$.

A square of area 25 has side 5, so $25^{\frac{1}{2}} = 5$: a power of a half is a square root. In general the bottom of the fraction is the root and the top is the power — $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 4$.

Pull out the largest square: $\sqrt{75} = 5\sqrt{3}$ and $\sqrt{27} = 3\sqrt{3}$. Now they're like terms — five lengths of $\sqrt{3}$ and three more — so $\sqrt{75} + \sqrt{27} = 8\sqrt{3}$.

Multiply by the conjugate and the difference of two squares wipes the surd out of the denominator: $\tfrac{6}{\sqrt{7} + 1} = \sqrt{7} - 1$. Same number — both are $1.64575\ldots$ — just with no surd underneath.

Count the tiles. Pull out the squares.

All six index laws, negative and fractional powers, harder conjugates, and twelve graded exam questions with full worked solutions are waiting.

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