Linear Algebra Examples

$27^{\frac{4}{3}} a \cdot 9^{- \frac{1}{2}} a \cdot 32^{\frac{2}{5}}$

Step 1

Convert expressions with fractional exponents to radicals.

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Step 1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\sqrt[3]{27^{4}} a \cdot 9^{- \frac{1}{2}} a \cdot 32^{\frac{2}{5}}$

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\sqrt[3]{27^{4}} a \cdot \frac{1}{9^{\frac{1}{2}}} a \cdot 32^{\frac{2}{5}}$

Step 1.3

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\sqrt[3]{27^{4}} a \cdot \frac{1}{\sqrt{9^{1}}} a \cdot 32^{\frac{2}{5}}$

Step 1.4

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\sqrt[3]{27^{4}} a \cdot \frac{1}{\sqrt{9^{1}}} a \cdot \sqrt[5]{32^{2}}$

Step 1.5

Anything raised to $1$ is the base itself.

$\sqrt[3]{27^{4}} a \cdot \frac{1}{\sqrt{9}} a \cdot \sqrt[5]{32^{2}}$

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3