Problem 79 - Precalculus: Functions and Graphs

Question

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 \(-3tv^2\).

Step 1: Apply the properties of radicals

Combine the two cube root expressions into a single cube root:\[\sqrt[3]{3t^4v^2} \cdot \sqrt[3]{-9t^{-1}v^4} = \sqrt[3]{(3t^4v^2) \times (-9t^{-1}v^4)}\].

Step 2: Simplify the expression under the cube root

Multiply the coefficients and the variables separately:- Coefficients: \(3 \times (-9) = -27\)- Variable \(t\): \(t^4 \times t^{-1} = t^{4-1} = t^3\)- Variable \(v\): \(v^2 \times v^4 = v^{2+4} = v^6\).Thus, the expression is:\[\sqrt[3]{-27t^3v^6}\].

Step 3: Simplify the cube root

Break down the expression within the cube root:\[\sqrt[3]{-27t^3v^6} = \sqrt[3]{(-3)^3 \times t^3 \times (v^2)^3}\].Since each part is a perfect cube, take each cube root individually:\[-3 \cdot t \cdot v^2 = -3tv^2\].

Key Concepts

Simplifying Expressions

Simplifying expressions is a vital skill in algebra. It involves condensing expressions into their simplest form while maintaining their original value. By doing this, calculations become easier and more straightforward.
To simplify an expression with multiple components, first combine like terms, which are terms that contain the same variables raised to the same power. Then, apply arithmetic operations such as addition, subtraction, multiplication, and division to reduce the expression further.
In our example, the terms inside the cube roots needed to be multiplied and simplified. By handling the coefficients and variables separately, we were able to simplify the expression under the cube root easily.

Cube Roots

The cube root is an important mathematical concept that pertains to finding a number which, when multiplied by itself three times, equals the original number. It is represented by the symbol \( \sqrt[3]{} \).
For example, the cube root of 27 is 3, since \(3 \times 3 \times 3 = 27\). Unlike square roots, cube roots can be applied to negative numbers because when a negative number is cubed, it remains negative.
In the exercise, we dealt with a cube root expression \( \sqrt[3]{-27t^3v^6} \). Taking the cube root of \(-27\), \(t^3\), and \(v^6\) individually gave simple results, which were then combined to provide the simplified form.

Rationalizing the Denominator

Rationalizing the denominator involves removing any radicals (such as square roots or cube roots) from the denominator of a fraction. Although this concept applies mainly to fractions, it is necessary here to ensure a clean final form of mathematical expressions.
This technique makes it easier to perform arithmetic operations and comparisons, as rational numbers are simpler to work with than their irrational counterparts.

• Multiply the numerator and denominator by a radical that will eliminate the radical in the denominator.
• The aim is to convert the denominator into a perfect power, allowing you to simplify the expression easily.

While our original exercise does not involve a fraction, the simplification of the cube root essentially operates under the same principle of making expressions easier to handle.

Properties of Exponents

Understanding the properties of exponents is crucial when simplifying expressions, particularly those involving multiplication and division of like bases with exponents.
Some basic properties include:

• Product of powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of powers: \(a^m / a^n = a^{m-n}\)
• Power of a power: \((a^m)^n = a^{mn}\)

These properties allow us to simplify expressions by combining and reducing terms.
In our solution, we applied these properties to variables \(t\) and \(v\), combining their exponents to achieve a simpler form of \(t^3\) and \(v^6\), which further simplified the cube root expression.