Convert to Radical Form 27^(-2/3)
The question asks for the conversion of the given exponential expression, which is 27 raised to the power of negative two-thirds (27^(-2/3)), into its equivalent radical form. This involves expressing the number using a root (such as a square root, cube root, etc.) instead of an exponent. The conversion process will utilize the properties of exponents and radicals to rewrite the given exponential term in a way that does not contain a fractional exponent.
$\left(27\right)^{\frac{- 2}{3}}$
Answer
Solution:
Step 1:
Consider the negative exponent in the expression $\left(27\right)^{-\frac{2}{3}}$.
Step 2:
Apply the rule for negative exponents: $a^{-m} = \frac{1}{a^m}$ to obtain $\frac{1}{\left(27\right)^{\frac{2}{3}}}$.
Step 3:
Transform the fractional exponent into a radical by using the rule $a^{\frac{m}{n}} = \sqrt[n]{a^m}$, resulting in $\frac{1}{\sqrt[3]{27^2}}$.
Step 4:
The final expression can be presented in its exact radical form or as a decimal approximation.
- Exact Radical Form: $\frac{1}{\sqrt[3]{27^2}}$
- Decimal Form: (not provided)
Knowledge Notes:
Negative Exponents: The negative exponent rule states that for any nonzero number $b$ and positive integer $n$, $b^{-n} = \frac{1}{b^n}$. This rule is an extension of the definition of exponentiation and is used to simplify expressions with negative exponents.
Fractional Exponents: A fractional exponent represents both an exponent and a root. The expression $x^{\frac{m}{n}}$ is equivalent to the nth root of $x$ raised to the mth power, denoted as $\sqrt[n]{x^m}$. This is particularly useful when converting between radical and exponential forms.
Radicals: A radical expression involves roots, such as square roots, cube roots, etc. The general form of a radical is $\sqrt[n]{x}$, where $n$ is the index of the root and $x$ is the radicand. Simplifying radical expressions often involves finding perfect nth powers that factor out of the radicand.
Simplifying Expressions: The process of simplifying expressions with exponents and radicals often involves applying exponent rules, such as the product of powers rule, the power of a power rule, and the power of a product rule, as well as simplifying radicals by extracting perfect nth powers.
Exact vs. Decimal Form: When expressing the result of a calculation, the exact form typically involves radicals and fractions, which represent the precise value. The decimal form is a numerical approximation of the exact value, which can be useful for practical purposes but may involve rounding.
