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The expression [tex]\frac{\left(27 y^{-2}\right)^{\frac{1}{3}}}{y^{-\frac{1}{3}}}[/tex] can be simplified and written without negative exponents as [tex]\square[/tex].


Sagot :

To simplify the expression [tex]\(\frac{\left(27 y^{-2}\right)^{\frac{1}{3}}}{y^{-\frac{1}{3}}}\)[/tex], follow these steps:

1. Let's start with the numerator [tex]\((27 y^{-2})^{1/3}\)[/tex].

- [tex]\(27\)[/tex] can be expressed as [tex]\(3^3\)[/tex], thus:
[tex]\[ (27 y^{-2})^{1/3} = \left( 3^3 y^{-2} \right)^{1/3} \][/tex]

- Using the power rule [tex]\((a^m \cdot b^n)^{k} = a^{mk} \cdot b^{nk}\)[/tex], we get:
[tex]\[ (3^3 y^{-2})^{1/3} = 3^{3 \cdot 1/3} \cdot y^{-2 \cdot 1/3} = 3 \cdot y^{-2/3} \][/tex]

2. Now, let's look at the denominator [tex]\(y^{-1/3}\)[/tex].

3. Combining the simplified numerator and denominator, we have:
[tex]\[ \frac{3 y^{-2/3}}{y^{-1/3}} \][/tex]

4. To simplify this division of exponents, recall the property of exponents [tex]\(a^{m}/a^{n} = a^{m-n}\)[/tex]:

- Thus, we can write:
[tex]\[ 3 \cdot y^{(-2/3) - (-1/3)} = 3 \cdot y^{(-2/3 + 1/3)} = 3 \cdot y^{-1/3} \][/tex]

So, the expression [tex]\(\frac{\left(27 y^{-2}\right)^{\frac{1}{3}}}{y^{-\frac{1}{3}}}\)[/tex] simplifies to:

[tex]\[ 3y^{-1/3} \][/tex]

To express the final answer in the required form without negative exponents:

[tex]\[ 3 \cdot y^{1/3} \][/tex]

The simplified expression is [tex]\( 3 y^{1/3} \)[/tex].