Convert to Radical Form 27^(-2/3)

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.