What is (4^{-2}*4^{-3})^3 ?

The solution to (4^{-2}*4^{-3})^3 is 1/1073741824

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simplify (4^{-2}*4^{-3})^3

simplify

(4^{- 2} \cdot 4^{- 3})^{3}

\frac{1}{1073741824}

Solution steps

(4^{- 2} \cdot 4^{- 3})^{3}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(4^{- 2} \cdot 4^{- 3})^{3} = (4^{- 2})^{3} (4^{- 3})^{3}

= (4^{- 2})^{3} (4^{- 3})^{3}

(4^{- 2})^{3} = \frac{1}{4096}

= \frac{1}{4096} (4^{- 3})^{3}

Apply exponent rule:

(a^{b})^{c} = a^{b c}, a \geq 0

(4^{- 3})^{3} = 4^{- 3 \cdot 3}

= \frac{1}{4096} \cdot 4^{- 3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{1}{4096} \cdot 4^{- 9}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

4^{- 9} = \frac{1}{4 ^{9}}

= \frac{1}{4096} \cdot \frac{1}{4 ^{9}}

4^{9} = 262144

= \frac{1}{4096} \cdot \frac{1}{262144}

\frac{1}{4096} \cdot \frac{1}{262144} = \frac{1}{1073741824}

= \frac{1}{1073741824}

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