Problem 22 - Algebra A Combined Function

Question

Use radical notation to rewrite each expression. Simplify if possible. $$ (-9)^{3 / 2} $$

Step-by-Step Solution

Answer

(-9)^{3/2} = -27i

Step 1: Understand the Radical Notation

The expression \((-9)^{3/2}\) can be broken down using the property \((a^{m/n} = \sqrt[n]{a^m})\). This means we can express \((-9)^{3/2}\) as a radical expression.

Step 2: Apply the Radical Conversion

Rewriting \((-9)^{3/2}\) as a radical expression involves breaking it into \((\sqrt{-9})^3\), because \((3/2 = 1/2 * 3)\). This is because the denominator 2 suggests a square root, and exponentiation follows.

Step 3: Simplify the Square Root

Evaluate \(\sqrt{-9}\). The square root of a negative number is not a real number, but for simplification in terms of imaginary numbers, \(-9\) can be written as \(-1 \cdot 9\). Thus, \(-1\)'s square root gives \(i\) and the square root of \(9\) is \(3\). Therefore, \(\sqrt{-9} = 3i\).

Step 4: Raise to the Power of 3

Now, cube \(3i\) as in \( (3i)^3 = 27i^3 \). Remember that \(i^3 = i^2 \cdot i = -i\) (since \(i^2 = -1\)), which means \(27i^3 = -27i\).

Step 5: Conclusion

The expression \((-9)^{3/2}\) when expressed in radical notation and simplified, is equal to \(-27i\).

Key Concepts

Imaginary Numbers

Imaginary numbers are a unique concept in mathematics. They arise when you need to take the square root of a negative number, which isn't possible with real numbers alone. The imaginary unit is denoted as \(i\) and is defined by the property \(i^2 = -1\). This means that the square root of \(-1\) is \(i\).

When dealing with negative numbers under a square root, such as \(\sqrt{-9}\), you break down the number into a product of \(-1\) and its positive counterpart, like \(-9 = -1 \cdot 9\). Hence, \(\sqrt{-9} = \sqrt{-1 \cdot 9} = \sqrt{-1} \cdot \sqrt{9} = i \cdot 3 = 3i\).

In essence, imaginary numbers extend our number system, allowing us to work with and solve equations that involve square roots of negative numbers. This is crucial in many fields, such as engineering and physics, where complex numbers play a significant role.

Simplifying Expressions

Simplifying expressions often involves reducing them to their most basic form while maintaining equivalence. In the context of the expression \((-9)^{3/2}\), simplifying involves breaking it down into more straightforward components and expressions.

Using radical notation, we see that the fraction in the exponent indicates a root and a power. The expression is decomposed using the identity \(a^{m/n} = \sqrt[n]{a^m}\), which in this case is \((-9)^{3/2} = (\sqrt{-9})^3\).

Through further simplification, we find that \(\sqrt{-9} = 3i\). So, the problem simplifies into cubing \(3i\): \((3i)^3\).

In mathematics, simplifying helps verify cumulative calculations, ensuring that no unnecessary complexity remains in an expression. It usually involves several steps such as factoring, combining like terms, and operating within established algebraic boundaries.

Exponentiation Rules

Exponentiation rules are fundamental laws that govern the operations involving powers of numbers. These rules help systematically handle expressions involving powers and roots, ensuring calculations' accuracy.

One critical rule is that when you raise a number to a fractional exponent, you involve both a root and another power: \((a^{m/n} = \sqrt[n]{a^m})\). This principle guided the transformation of \((-9)^{3/2}\) into \((\sqrt{-9})^3\).

Another essential rule is the multiplication of exponents: \((a^m)^n = a^{m\cdot n}\). For our expression, once \(\sqrt{-9}\) was resolved to \(3i\), cubing it follows the exponentiation property: \((3i)^3 = 27i^3\).

Furthermore, understanding \(i^3 = i^2 \times i = -i\) plays a vital role. Knowing these rules allows you to perform complex calculations with confidence, ensuring that each step adheres to mathematical conventions.