Problem 49 - Algebra 2

Question

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \frac{10}{\sqrt[3]{5 x^{2}}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \( \frac{10}{\sqrt[3]{5x^{2}}} \) is \( 2*5^{\frac{2}{3}}*x^{\frac{-2}{3}} \)

Step 1: Understand the problem

Here, the task is to simplify the expression \(\frac{10}{\sqrt[3]{5x^{2}}}\) and rationalize the denominator. We want to eliminate the cube root in the denominator.

Step 2: Rationalize the denominator

In order to rationalize the denominator, we can multiply and divide by the cube of \(\sqrt[3]{5x^{2}}\). This gives us: \(\frac{10}{\sqrt[3]{5x^{2}}}\) * \(\frac{\sqrt[3]{5x^{2}}^2}{\sqrt[3]{5x^{2}}^2}\) = \(\frac{10\sqrt[3]{5x^{2}}^2}{\sqrt[3]{5x^{2}}^3}\)

Step 3: Simplify the expression

Simplify the above expression, using that \( \sqrt[3]{5x^{2}}^3=5x^{2}\) in the denominator and power of power rule in the numerator. So it simplifiesto \(\frac{10*(5x^2)^{\frac{2}{3}}}{5x^{2}}\) = \(2*(5x^2)^{\frac{2}{3}}/x^{2}\)

Step 4: Simplify further

Finally simplify the expression to it's simplest form. This simplifies to \(2*5^{\frac{2}{3}}*x^{\frac{4}{3}}/x^{2}\) = \(2*5^{\frac{2}{3}}*x^{\frac{4}{3}-2}\) = \(2*5^{\frac{2}{3}}*x^{\frac{-2}{3}}\). This is the simplified expression with the denominator rationalized.

Key Concepts

Cube Roots

Cube roots are operations used to determine a number that, when multiplied by itself thrice, gives the original number. The cube root of a number is symbolized as \( \sqrt[3]{x} \). It's helpful for understanding the structure of numbers, especially when they appear in denominators of expressions.

• Consider \( \sqrt[3]{27} \). This indicates what number multiplied by itself three times equals 27. The answer is 3, since \( 3 \times 3 \times 3 = 27 \).
• In algebraic expressions, cube roots can include variables, like \( \sqrt[3]{8x^3} \). Here, both the numeric and variable parts undergo cube rooting, resulting in \( 2x \), since \( 2x \times 2x \times 2x = 8x^3 \).

Recognizing cube roots is pivotal in manipulating and understanding algebraic expressions, especially when simplifying terms or rationalizing denominators.

Simplifying Expressions

Simplifying an expression involves reducing it to its most basic form while keeping its value intact. It often means eliminating unnecessary components like radicals or fractional exponents.

• Consider the fraction \( \frac{10}{\sqrt[3]{5x^2}} \). The goal is to simplify this by removing the cube root from the denominator.
• This is done by multiplying both the numerator and denominator by \( \sqrt[3]{(5x^2)^2} \), essentially translating the cube root into something manageable.

Simplification often involves recognizing powers of power rules, which dictate that when you raise a power to another power, you multiply the exponents, such as \( (x^m)^n = x^{m\cdot n} \). This principle is crucial when simplifying fractions with radicals.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. These expressions need to be manipulated according to algebra rules to simplify or alter their form.

• Variables, like \( x \), represent unknowns and can take different values, making expressions versatile.
• To transform expressions, especially those with roots or fractions, you can use several algebraic techniques including multiplication, division, and use of exponents.

For the given expression \( \frac{10}{\sqrt[3]{5x^2}} \), algebraic manipulation involves rationalizing the denominator. By converting \( \sqrt[3]{5x^2} \) to a form without a radical, you align it more closely with other powers, enabling easier operations with the expression.

• The final form of such expressions, like \( 2 \times 5^{\frac{2}{3}} \times x^{\frac{-2}{3}} \), involves understanding how negative exponents indicate reciprocal forms and fractional exponents relate to roots.

Mastering these forms makes working with complex algebraic expressions smoother and more intuitive.