Evaluate each expression.

[tex]\[ \left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}} = \frac{1}{5} \][/tex]

[tex]\[ \frac{27^2}{27^{\frac{4}{3}}} = \][/tex]



Answer :

Sure, let's work through the expression step by step to solve [tex]\(\frac{27^2}{27^{\frac{4}{3}}}\)[/tex].

### Step-by-Step Solution:

1. Write down the given expression:

[tex]\[ \frac{27^2}{27^{\frac{4}{3}}} \][/tex]

2. Apply the laws of exponents:

According to the laws of exponents, specifically the quotient rule [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], we can simplify this expression by subtracting the exponents:

[tex]\[ 27^2 \div 27^{\frac{4}{3}} = 27^{2 - \frac{4}{3}} \][/tex]

3. Simplify the exponent:

We need to subtract [tex]\(\frac{4}{3}\)[/tex] from [tex]\(2\)[/tex]. To do this, it's helpful to express [tex]\(2\)[/tex] with a common denominator:

[tex]\[ 2 = \frac{6}{3} \][/tex]

Now we perform the subtraction:

[tex]\[ \frac{6}{3} - \frac{4}{3} = \frac{6 - 4}{3} = \frac{2}{3} \][/tex]

So, the exponent simplifies to [tex]\(\frac{2}{3}\)[/tex].

4. Rewrite the expression:

Now that we have simplified the exponents, we get:

[tex]\[ 27^{\frac{2}{3}} \][/tex]

5. Evaluate [tex]\(27^{\frac{2}{3}}\)[/tex]:

To evaluate [tex]\(27^{\frac{2}{3}}\)[/tex], we understand that the exponent [tex]\(\frac{2}{3}\)[/tex] indicates taking the cube root of 27 and then squaring the result.

- The cube root of 27:

[tex]\(\sqrt[3]{27} = 3\)[/tex] because [tex]\(3^3 = 27\)[/tex].

- Then square this result:

[tex]\(3^2 = 9\)[/tex].

Therefore, the expression [tex]\(\frac{27^2}{27^{\frac{4}{3}}}\)[/tex] evaluates to [tex]\(9\)[/tex].

[tex]\(\boxed{9}\)[/tex]