Problem 79 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt[3]{3 t^{4} v^{2}} \sqrt[3]{-9 t^{-1} v^{4}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-3 t v^{2}\).

Step 1: Simplify Each Radicand

Start by simplifying each cube root individually. For the first radicand, we have \( \sqrt[3]{3 t^{4} v^{2}} \), which is just \( 3 t^{4} v^{2} \). The second radicand is \( \sqrt[3]{-9 t^{-1} v^{4}} \), which is \( -9 t^{-1} v^{4} \).

Step 2: Multiply the Radicands

Multiply the simplified expressions inside each cube root: \( (3 t^{4} v^{2}) \times (-9 t^{-1} v^{4}) = -27 t^{3} v^{6} \).

Step 3: Simplify the Cube Root of the Product

Take the cube root of the resulting product: \( \sqrt[3]{-27 t^{3} v^{6}} \). Since \(-27 = (-3)^3\), \(t^3 = (t)^3\), and \(v^6 = (v^2)^3\), the cube root simplifies to \(-3 t v^{2}\).

Step 4: State the Simplified Expression

Combine the results from previous steps. The expression simplifies to \(-3 t v^{2}\).

Key Concepts

Cube Roots

Understanding cube roots involves identifying a number that, when multiplied by itself twice, results in the original number. In mathematical terms, the cube root of a number \(a\) is a number \(b\) such that \(b^3 = a\).
Cube root simplifies handling expressions involving special powers. It's symbolized as \(\sqrt[3]{x}\). Let's break down the expression:

• Given the cube root \( \sqrt[3]{3 t^4 v^2} \), you leave the expression as it is since it's already under a cube root.
• Similarly, \( \sqrt[3]{-9 t^{-1} v^4} \) remains in its cube root form in the problem​.

Cube roots allow you to simplify products inside radicals by breaking them into more manageable parts, making further simplification easier.

Rationalizing Denominators

Rationalizing the denominator involves eliminating any radicals in the denominator of a fraction. While the initial problem doesn't directly involve fractions with cube roots in the denominator, this concept is crucial when cube roots appear in denominators. To rationalize denominators involving cube roots:

• You multiply both the numerator and the denominator by the appropriate factor to ensure the radical disappears from the denominator.
• For cube roots, specifically, multiply by the cube of a number to simplify and remove the radical from the denominator.

For instance, simplifying \(\frac{1}{\sqrt[3]{a}}\) involves multiplying both the numerator and the denominator by \(\sqrt[3]{a^2}\), since \(a \times a^2 = a^3\), which results in \(a\). Understanding how to rationalize cube roots prepares you for more complex algebra involving radicals.

Multiplying Expressions

Multiplying expressions involves combining like terms and simplifying the product. For the expression \(\sqrt[3]{3 t^4 v^2} \times \sqrt[3]{-9 t^{-1} v^4}\), start by multiplying the contents inside the cube roots.

• The general rule is to multiply coefficients together, which in this case is \(3\times(-9) = -27\).
• Add the exponents of like bases, giving \(t^{4+(-1)}=t^3\).
• For \(v\), add the exponents: \(v^{2+4}=v^6\).

So, multiplying inside the roots results in \(-27t^3v^6\). Multiplying expressions allows for simplification and further breaking down complex expressions into easier-to-manage forms.

Polynomial Expressions

Polynomials are mathematical expressions that involve a sum of powers in one or more variables multiplied by coefficients. Each separate term in a polynomial is a monomial, which consists of a coefficient and variables raised to non-negative integers.
In our exercise, the expression "\(-27t^3v^6\)" from multiplication of cube root expressions behaves like a polynomial, although it is a single term.

• When simplifying polynomials under radicals, the key is simplifying each factor's power separately.
• Once simplified from cube roots to whole numbers (like \(-3t v^2\)), it becomes easier to manage and understand.

This expression's simplification into a polynomial expression allows manipulation through addition, subtraction, and further simplifications. Understanding polynomial expressions helps in rewriting and reorganizing expressions to uncover simpler forms.