Problem 49 Solve. $$ 8 x^{3}=27 $$... [FREE SOLUTION]

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Chapter 6: Problem 49

Solve. $$ 8 x^{3}=27 $$

Short Answer

Expert verified

x = \frac{3}{2}

Step by step solution

Identify the Equation Type

Recognize that the given equation is a cubic equation: $8x^{3}=27$.

Isolate the Cubic Term

Divide both sides of the equation by 8 to isolate the cubic term: $x^3 = \frac{27}{8}$.

Take the Cube Root of Both Sides

Find the cube root of both sides to solve for $x$: $x = \sqrt[3]{\frac{27}{8}}$.

Simplify the Cube Root Expression

Simplify $x = \sqrt[3]{\frac{27}{8}}$ using the property of cube roots: $x = \sqrt[3]{27} / \sqrt[3]{8}= 3/2$.

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Key Concepts

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

headline of the respective core concept

Cubic equations are a type of polynomial equation where the highest degree term is raised to the power of three. In this exercise, we are dealing with the equation: \[ 8x^{3} = 27 \].Cubic equations can often appear complex, but breaking them down into simpler steps can make solving them more manageable. One method to solve them is by isolating the cubic term and then taking the cube root. This step-by-step approach will help you understand how to solve such equations efficiently.

Cube Roots

To solve a cubic equation, taking the cube root of both sides is a critical step. When isolating the cubic term, as in the example \[ 8x^{3} = 27 \], we focus on getting $ x^{3} $ by itself first: \[ x^{3} = \frac{27}{8} \].Next, we find the cube root of both sides. The cube root of a number $ a $ is the value $ b $such that $ b^{3} = a $. In this case, we need to determine $ \sqrt[3]{\frac{27}{8}} $. You can break it down further: \[ x = \frac{\sqrt[3]{27}}{\sqrt[3]{8}} \]. Once simplified, this becomes: \[ x = \frac{3}{2} \].

Algebraic Simplification

Algebraic simplification makes complex expressions easier to manage by reducing them to simpler forms. When simplifying cube roots, the properties of exponents and roots can be utilized. For instance, the equation $ 8x^3 = 27 $ can be transformed step-by-step:

First, isolate the term $ x^{3} $.
Second, take the cube root of both sides: $ x = \sqrt[3]{\frac{27}{8}} $.

Recognize the cube roots: $ x = \frac{\sqrt[3]{27}}{\sqrt[3]{8}} $.Since $ 27 = 3^3 $ and $ 8 = 2^3 $, we get: $ x = \frac{3}{2} $.Practicing these steps helps in gaining a better understanding of how to simplify similar algebraic expressions. This process not only applies to cubic equations but enhances your overall problem-solving skills in algebra.

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91影视

Solve. $$ 8 x^{3}=27 $$

Short Answer

Step by step solution

Identify the Equation Type

Isolate the Cubic Term

Take the Cube Root of Both Sides

Simplify the Cube Root Expression

Key Concepts

headline of the respective core concept

Cube Roots

Algebraic Simplification

One App. One Place for Learning.

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