Exponent rules

$x^{5}$ just means five $x$ 's multiplied together. Every exponent rule follows from counting them, so you don't need to memorize blindly.

Work through it

Write the factors.three $x$ 's, then four more

Count them. $x^{7}$

Square both parts. $\frac{4 x ^{6}}{x ^{2}}$

Subtract exponents. $4 x^{4}$

Each step down divides by 5: $5^{1} = 5$ , $5^{0} = 1$ . $5^{- 1} = \frac{1}{5}$

One more. $5^{- 2} = \frac{1}{25}$

$x^{3} \cdot x^{4} = x^{12}$ .

That's multiplying exponents. Count the factors: 7. And $(x + y)^{2}$ is not $x^{2} + y^{2}$ .

The short version
Exponents count factors. Multiplying adds exponents, dividing subtracts, and a power of a power multiplies.

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