Simplify the following expression 3^11/5 divided by 3^-9/5
The correct Answer and Explanation is:
The given expression is:
[
\frac{3^{11/5}}{3^{-9/5}}
]
To simplify this, we can use the properties of exponents. One key property that applies here is:
[
\frac{a^m}{a^n} = a^{m-n}
]
Using this property, we can rewrite the expression as:
[
3^{\frac{11}{5} – \left(-\frac{9}{5}\right)}
]
Simplifying the exponent subtraction:
[
\frac{11}{5} – (-\frac{9}{5}) = \frac{11}{5} + \frac{9}{5} = \frac{11 + 9}{5} = \frac{20}{5} = 4
]
Now the expression becomes:
[
3^4
]
The value of (3^4) is:
[
3^4 = 3 \times 3 \times 3 \times 3 = 81
]
Thus, the simplified expression is:
[
81
]
Explanation
In this problem, we were tasked with simplifying an expression involving exponents. The expression was (\frac{3^{11/5}}{3^{-9/5}}), which requires applying the rule for dividing exponents with the same base. When dividing terms with the same base (in this case, base 3), we subtract the exponents.
The exponents in the expression were (11/5) and (-9/5). By subtracting (-9/5) from (11/5), we end up adding (9/5) because subtracting a negative is the same as adding a positive. The result of this addition is (20/5), which simplifies to 4.
Once we have simplified the exponent to 4, the expression becomes (3^4). This means we multiply 3 by itself four times, which gives 81.
Understanding these rules of exponents is important because they allow us to work with very large or very small numbers efficiently without needing to write out long calculations. This skill is essential in many areas, including algebra, calculus, and real-world applications like computing compound interest or growth rates.