(3/2) to the 4th power
The Correct Answer and Explanation is :
To calculate (\left(\frac{3}{2}\right)^4), we need to apply the rules of exponents to fractional numbers. The expression (\left(\frac{3}{2}\right)^4) means we multiply the fraction (\frac{3}{2}) by itself four times.
Step-by-Step Calculation:
[
\left(\frac{3}{2}\right)^4 = \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2}
]
This can be simplified by applying the rule of exponents for fractions, which states that:
[
\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
]
So, we apply this rule to (\left(\frac{3}{2}\right)^4):
[
\left(\frac{3}{2}\right)^4 = \frac{3^4}{2^4}
]
Now we calculate (3^4) and (2^4):
- (3^4 = 3 \times 3 \times 3 \times 3 = 81)
- (2^4 = 2 \times 2 \times 2 \times 2 = 16)
Thus, we have:
[
\left(\frac{3}{2}\right)^4 = \frac{81}{16}
]
Final Answer:
[
\frac{81}{16}
]
Explanation of the Concept:
The process of raising a fraction to a power follows the same principles as raising any number to a power, but with the added step of separately applying the exponent to the numerator and denominator. The rule of exponents for fractions allows us to handle fractional numbers in an organized manner by raising both the numerator and denominator to the same power.
Exponentiation involves repeated multiplication. For a fraction (\frac{a}{b}), the exponent tells us how many times to multiply the fraction by itself. By separating the powers of the numerator and the denominator, we maintain the integrity of the fraction’s form and ensure the calculations are manageable.
Thus, (\left(\frac{3}{2}\right)^4 = \frac{81}{16}) is the correct and simplified result.