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QuestionSimplify the expression (aβ3)(a−2β4)5\left(a \beta^{3}\right)\left(a^{-2} \beta^{4}\right)^{5}.

Studdy Solution

STEP 1

Assumptions1. The given expression is (aβ3)(a−β4)5\left(a \beta^{3}\right)\left(a^{-} \beta^{4}\right)^{5} . The properties of exponents apply, including - am⋅an=am+na^{m} \cdot a^{n} = a^{m+n} - (am)n=am⋅n(a^{m})^{n} = a^{m \cdot n}

STEP 2

First, we need to simplify the expression inside the parentheses. We can do this by using the property of exponents am⋅an=am+na^{m} \cdot a^{n} = a^{m+n}.
(a−2β4)5=a−2⋅5⋅β4⋅5\left(a^{-2} \beta^{4}\right)^{5} = a^{-2 \cdot5} \cdot \beta^{4 \cdot5}

STEP 3

Now, calculate the exponents.
(a−2β)5=a−10⋅β20\left(a^{-2} \beta^{}\right)^{5} = a^{-10} \cdot \beta^{20}

STEP 4

Substitute the simplified expression back into the original equation.
(aβ3)(a−10β20)\left(a \beta^{3}\right)\left(a^{-10} \beta^{20}\right)

STEP 5

Again, use the property of exponents am⋅an=am+na^{m} \cdot a^{n} = a^{m+n} to simplify the equation.
a1⋅a−10⋅β3⋅β20=a1−10⋅β3+20a^{1} \cdot a^{-10} \cdot \beta^{3} \cdot \beta^{20} = a^{1-10} \cdot \beta^{3+20}

STEP 6

Calculate the exponents.
a1−10⋅β3+20=a−9⋅β23a^{1-10} \cdot \beta^{3+20} = a^{-9} \cdot \beta^{23}So, (aβ3)(a−2β4)5=a−9⋅β23\left(a \beta^{3}\right)\left(a^{-2} \beta^{4}\right)^{5} = a^{-9} \cdot \beta^{23}.

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