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Chapter 10: Problem 64

Divide and, if possible, simplify. $$\frac{\sqrt[3]{x^{2}+7 x+12}}{\sqrt[3]{x+3}}$$

Short Answer

Expert verified
The simplified form of the expression \(\frac{\sqrt[3]{x^{2}+7 x+12}}{\sqrt[3]{x+3}}\) is \(\sqrt[3]{x+4}\).

Step by step solution

01

Factor the numerator polynomial

First, factor the polynomial in the numerator of the fraction. \(x^{2}+7x+12\) can be factored into \((x+3)(x+4)\). So now we have \(\frac{\sqrt[3]{(x+3)(x+4)}}{\sqrt[3]{x+3}}\).
02

Split the Cube Root

Next, split the cube root in the numerator into separate cube roots for each factor to get \(\frac{\sqrt[3]{x+3} \cdot \sqrt[3]{x+4}}{\sqrt[3]{x+3}}\).
03

Simplify the Expression

In this step, we will simplify the expression. We can cancel out the common \(\sqrt[3]{x+3}\) term from the numerator and denominator. So, what we are left with is \(\sqrt[3]{x+4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Polynomials
Factoring polynomials is a crucial skill in algebra that allows us to break down complex expressions into simpler, more manageable parts, often making further calculations or simplifications possible.

Consider the polynomial expression in the given exercise, \( x^{2}+7x+12 \). To factor this, we look for two numbers that add up to 7 (the coefficient of the x term) and multiply to 12 (the constant term). In this case, those numbers are 3 and 4, allowing us to rewrite the quadratic as \( (x+3)(x+4) \). Factoring makes it easier to simplify expressions, especially when dealing with rational expressions involving radicals, as seen in our exercise.
Radical Expressions
A radical expression involves roots, such as square roots, cube roots, and higher. Simplifying radical expressions can often involve factoring polynomials, as we've seen in the exercise.

It's important to understand that \( \sqrt[n]{ab} \sim \sqrt[n]{a}\cdot\sqrt[n]{b} \), where 'n' is the index of the root and 'a' and 'b' are terms within the radical. In the context of the exercise given, separating the cube root of the factored polynomial into the product of two cube roots makes it possible to cancel out terms and simplify the problem. Mastery of radical expressions is vital for higher-level mathematics, where such expressions frequently occur.
Algebraic Simplification
Algebraic simplification involves reducing expressions to their simplest form. This can be done through various methods, including canceling out common factors, simplifying radical expressions, and combining like terms.

In our exercise, algebraic simplification comes into play when the common \( \sqrt[3]{x+3} \) is identified in both the numerator and the denominator, enabling us to cancel this term from the expression. The result, \( \sqrt[3]{x+4} \), is a much simpler and more elegant form, showcasing the power of algebraic simplification in making expressions clearer and more digestible.

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Most popular questions from this chapter

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{3 \sqrt{x}+\sqrt{y}}{\sqrt{y}-3 \sqrt{x}}$$

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{2 \sqrt{6}+\sqrt{5}}{3 \sqrt{6}-\sqrt{5}}$$

In solving \(\sqrt{2 x-1}+2=x,\) why is it a good idea to isolate the radical term? What if we don't do this and simply square each side? Describe what happens.

In Exercises \(105-112,\) add or subtract as indicated. Begin by rationalizing denominators for all terms in which denominators contain radicals. $$\sqrt{15}-\sqrt{\frac{5}{3}}+\sqrt{\frac{3}{5}}$$

Solve each equation. $$(x-4)^{\frac{2}{3}}=25$$
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