To simplify the expression [tex]\( 81^{\frac{5}{4}} \)[/tex]:
1. Recognize the form of the expression:
The expression is given in the form [tex]\( a^{\frac{m}{n}} \)[/tex], where [tex]\( a = 81 \)[/tex], [tex]\( m = 5 \)[/tex], and [tex]\( n = 4 \)[/tex].
2. Rewrite the base as a power of its prime factors:
Notice that [tex]\( 81 \)[/tex] can be expressed as [tex]\( 3^4 \)[/tex].
Therefore, [tex]\( 81 = 3^4 \)[/tex].
3. Substitute this into the original expression:
Substitute [tex]\( 81 \)[/tex] into the expression you get:
[tex]\[
81^{\frac{5}{4}} = (3^4)^{\frac{5}{4}}
\][/tex]
4. Simplify the exponent:
Use the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex].
Here, [tex]\((3^4)^{\frac{5}{4}} = 3^{4 \cdot \frac{5}{4}} = 3^5\)[/tex].
5. Evaluate the exponentiation:
[tex]\( 3^5 \)[/tex] means [tex]\( 3 \times 3 \times 3 \times 3 \times 3 \)[/tex].
6. Calculate the result of [tex]\( 3^5 \)[/tex]:
When you perform this multiplication:
[tex]\[
3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243
\][/tex]
Thus, [tex]\( 81^{\frac{5}{4}} \)[/tex] simplifies to [tex]\( 243 \)[/tex]. The correct answer is [tex]\( 243 \)[/tex].
