Law of multiplication of indices

a^m x aⁿ = a^{m + m}

Law of division of indices

$\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^{m - n}

a⁰ = 1

$\frac{5^{7}x{5}^{-4}}{5^3}$ = $\frac{5^{\mathrm{7+-4}}}{5^3}$

$\frac{5^{7}x{5}^{-4}}{5^3}$ = $\frac{5^{\mathrm{7-4}}}{5^3}$

$\frac{5^{7}x{5}^{-4}}{5^3}$ = $\frac{5^3}{5^3}$ = 1

Note: 5³ divides 5³ once.

Alternatively, the law of division could be applied.

$\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^{m - n}

$\frac{5^3}{5^3}$ = 5^3-3

$\frac{5^3}{5^3}$ = 5⁰ = 1

5⁰ = 1 (a⁰ = 1)

Convert the 81 to 3^y, where y is the power that will produce 81

81 = 3 x 3 x 3 x 3

From the law of indices, a^b x a^c = a^b+c

81 = 3 x 3 x 3 x 3 = 3^1+1+1+1 = 3⁴

Now replace 81 by 3⁴ in the equation

3^x = 81

3^x = 3⁴

Since the bases are the same, the powers are also the same

Hence x = 4

Applying the indices principle

The expression can therefore be rewritten as:

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