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Exponents – Running Notes
17

Exponents (Level 1)

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Worked Examples Part 3

Example 8

Simplify:

$\left(\frac{1}{2}\right)^{-2} + \left(\frac{1}{3}\right)^{-2} + \left(\frac{1}{4}\right)^{-2}$

Using negative exponent law:

$\left(\frac{1}{2}\right)^{-2} = 2^2 = 4$

$\left(\frac{1}{3}\right)^{-2} = 3^2 = 9$

$\left(\frac{1}{4}\right)^{-2} = 4^2 = 16$

So,

$4 + 9 + 16 = 29$

Therefore,

$\left(\frac{1}{2}\right)^{-2} + \left(\frac{1}{3}\right)^{-2} + \left(\frac{1}{4}\right)^{-2} = 29$

Example 9

Simplify:

$(-1)^2 + (-1)^3 + (-1)^5$

$(-1)^2 = 1$

$(-1)^3 = -1$

$(-1)^5 = -1$

So,

$1 - 1 - 1 = -1$

Therefore,

$(-1)^2 + (-1)^3 + (-1)^5 = -1$

Example 10

Simplify:

$(-2)^6 + (-4)^3$

$(-2)^6 = 2^6 = 64$

$(-4)^3 = -4^3 = -64$

So,

$64 - 64 = 0$

Therefore,

$(-2)^6 + (-4)^3 = 0$

Example 11

If

$5^{3m+1} = \frac{1}{125}$

find $m$ .

We know:

$125 = 5^3$

So,

$\frac{1}{125} = \frac{1}{5^3} = 5^{-3}$

Given:

$5^{3m+1} = \frac{1}{125}$

Therefore,

$5^{3m+1} = 5^{-3}$

Since the bases are equal,

$3m + 1 = -3$

$3m = -3 - 1$

$3m = -4$

$m = -\frac{4}{3}$

Therefore,

$m = -\frac{4}{3}$

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