3/2 to the power of 3 as a fraction
The Correct Answer and Explanation is :
The correct answer is: frac{27}{8}
To calculate ( \left(\frac{3}{2}\right)^3 ), we raise both the numerator and the denominator to the power of 3.
- Calculate the Numerator:
[
3^3 = 3 \times 3 \times 3 = 27
] - Calculate the Denominator:
[
2^3 = 2 \times 2 \times 2 = 8
]
Putting it all together, we get:
[
\left(\frac{3}{2}\right)^3 = \frac{27}{8}
]
Explanation:
When we deal with fractions and exponents, it’s essential to remember the basic rules of exponents. For any fraction ( \frac{a}{b} ) raised to the power of ( n ), the result can be expressed as:
[
\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
]
This means you raise both the numerator and denominator to the same exponent. In this case, ( a = 3 ), ( b = 2 ), and ( n = 3 ).
Raising the numerator and the denominator separately simplifies the calculation and helps to maintain accuracy.
Converting to Mixed Numbers:
The fraction ( \frac{27}{8} ) can also be converted into a mixed number. To do this, we divide 27 by 8.
[
27 \div 8 = 3 \quad \text{(with a remainder of 3)}
]
Thus, ( \frac{27}{8} ) can be expressed as:
[
3 \frac{3}{8}
]
Final Result:
So, the final answer to ( \left(\frac{3}{2}\right)^3 ) is ( \frac{27}{8} ), which can also be represented as the mixed number ( 3 \frac{3}{8} ). This demonstrates how exponentiation works with fractions and provides an understanding of how to manipulate and convert between improper fractions and mixed numbers.