Simplify (1/(9^(3/2)))
The problem asks for the simplification of the fractional expression where the denominator is a power of the number 9, specifically raised to the power of three-halves. The expression needs to be simplified to its simplest form using the rules of exponents and possibly the properties of fractions.
$\left(\right. \frac{1}{9^{\frac{3}{2}}} \left.\right)$
Answer
Solution:
Step 1:
Express the given fraction without the parentheses: $\frac{1}{9^{\frac{3}{2}}}$.
Step 2:
Begin simplifying the expression in the denominator.
Step 2.1:
Express $9$ as a power of $3$: $\frac{1}{(3^{2})^{\frac{3}{2}}}$.
Step 2.2:
Utilize the property of exponents that $(a^{m})^{n} = a^{mn}$: $\frac{1}{3^{2 \cdot \frac{3}{2}}}$.
Step 2.3:
Simplify the exponent by reducing the fraction.
Step 2.3.1:
Reduce the fraction in the exponent: $\frac{1}{3^{\cancel{2} \cdot \frac{3}{\cancel{2}}}}$.
Step 2.3.2:
Simplify the expression to its final form: $\frac{1}{3^{3}}$.
Step 2.4:
Calculate the value of $3$ raised to the third power: $\frac{1}{27}$.
Step 3:
Present the final result in its various forms.
Exact Form: $\frac{1}{27}$
Decimal Form: $0.037037...$
Knowledge Notes:
To simplify an expression like $\frac{1}{9^{\frac{3}{2}}}$, one must understand the following concepts:
Exponentiation: This is a mathematical operation, involving two numbers, the base and the exponent. When a number is raised to a power, the base is multiplied by itself as many times as the value of the exponent.
Fractional Exponents: A fractional exponent represents both an exponent and a root. For example, $a^{\frac{1}{n}}$ is the nth root of a, and $a^{\frac{m}{n}}$ is the nth root of $a^m$.
Power of a Power Rule: This rule states that when you raise a power to another power, you multiply the exponents. In mathematical terms, $(a^m)^n = a^{mn}$.
Simplifying Fractions: When simplifying fractions, one can cancel out common factors in the numerator and the denominator to reduce the fraction to its simplest form.
Rationalizing the Denominator: This involves rewriting an expression to remove radicals from the denominator. In this case, however, we are dealing with exponents rather than radicals.
Decimal Representation: Some fractions can be represented as repeating or terminating decimals. The fraction $\frac{1}{27}$, for instance, is a repeating decimal that can be approximated as 0.037.
By applying these principles, one can simplify the given expression step by step, ensuring that each operation follows the rules of arithmetic and algebra.
