Problem 34 - College Algebra with Modeling and Visualization

Question

If possible, simplify the expression by hand. If you cannot, approximate the answer to the nearest hundredth. Variables represent any real number. $$ \sqrt[3]{(x+1)^{6}} $$

Step-by-Step Solution

Answer

The simplified expression is \((x+1)^2\).

Step 1: Understand the Expression

The given expression is \( \sqrt[3]{(x+1)^6} \). This is a cube root of \( (x+1)^6 \). Our goal is to simplify this expression, if possible.

Step 2: Use Property of Exponents

Recall the property \( \sqrt[n]{a^m} = a^{m/n} \). Applying this to our expression, we have \( (x+1)^{6/3} \).

Step 3: Simplify the Exponent

Simplify \( 6/3 \) to find the new power of the expression. \( 6/3 = 2 \), thus the expression now becomes \( (x+1)^2 \).

Step 4: Final Simplified Expression

The simplified expression is \( (x+1)^2 \). This cannot be simplified further without additional information about \( x \).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. In our expression, we have \((x+1)^6\), meaning the quantity \((x+1)\) is multiplied by itself 6 times. We use exponents in algebra to make complex expressions easier to write and solve.
Exponents follow specific rules that help in their manipulation:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Power of a Product: \((ab)^m = a^m \times b^m\)
• Quotient of Powers: \(a^m / a^n = a^{m-n}\)

Using these rules, we can simplify expressions and solve equations more efficiently.

Simplification

Simplifying an expression often involves using algebraic rules and properties to rewrite it in a more manageable or aesthetically pleasing form. In this exercise, we simplify \( \sqrt[3]{(x+1)^6} \) by manipulating the exponents. Simplification in algebra may involve combining like terms, reducing fractions within exponents, or using specific rules for operations.
In this expression, we apply the property \( \sqrt[n]{a^m} = a^{m/n} \), transforming \( (x+1)^6 \) under a cube root to \( (x+1)^{6/3} \). This exponent fraction \( 6/3 \) is then simplified to 2. Therefore, the expression becomes \( (x+1)^2 \).
Understanding how to manage and reduce complex expressions leads to greater mathematical accuracy and efficiency, especially in solving more complicated problems.

Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. Cube roots undo the cubic effect an exponent has when applied thrice. It’s commonly denoted as \( \sqrt[3]{x} \).
For \( \sqrt[3]{(x+1)^6} \), we seek a number that, when cubed, results in \((x+1)^6\). We can find this by using the exponent division property: essentially, dividing the exponent by 3. This process simplifies the cube root problem to finding \((x+1)^2\).
Cube roots are particularly useful in solving polynomial equations and understanding geometric scale transformations in various mathematics applications. When you master cube roots, you can tackle problems involving volumes or transformations with more ease.