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]
