Get the solution to "Simplify and rewrite the expression so each variable..."

Tobias Ali

Answered question

2021-09-13

Simplify and rewrite the expression so each variable only appears once.

\sqrt{a^{16} b^{10}} \cdot {(a^{- 2} b^{3})}^{4}

2k1enyvp

Skilled2021-09-14Added 94 answers

We can use the exponent rules,

a^{m} \cdot a^{n} = a^{m + n}

{(a^{m})}^{n} = a^{m n}

We can also use the radical to exponent conversion rules,

\sqrt{n} {a^{m}} = a^{\frac{m}{n}}

\sqrt{a^{n}} = a^{\frac{1}{n}}

Also we can use the radical form

\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}

Consider the expression

\sqrt{a^{16} b^{10}} \cdot {(a^{- 2} b^{3})}^{4}

Use the radical form

\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}

to simplify the radical term and split the exponent term.

\sqrt{a^{16} b^{10}} \cdot {(a^{- 2} b^{3})}^{4} = \sqrt{a^{16}} \cdot \sqrt{b^{10}} \cdot {(a^{- 2})}^{4} \cdot {(b^{3})}^{4}

To further simplify the expression use the rules,

{(a^{m})}^{n} = a^{m n}, \sqrt{n} {a^{m}} = a^{\frac{m}{n}}

, we get

\sqrt{a^{16}} \cdot \sqrt{b^{10}} \cdot {(a^{- 2})}^{4} \cdot {(b^{3})}^{4} = a^{\frac{16}{2}} \cdot b^{\frac{10}{2}} \cdot a^{- 2 (4)} \cdot b^{3 (4)}

= a^{8} \cdot b^{5} \cdot a^{- 8} \cdot b^{12}

= a^{8 - 8} \cdot b^{5 + 12}

= a^{0} \cdot b^{17}

= 1 \cdot b^{17}

= b^{17}

Hence the simplest form of the given expression is

b^{17}

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