Problem 71 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt[3]{x}+1)(\sqrt[3]{x}-4 \sqrt{x}+7) $$

Step-by-Step Solution

Answer

The simplified expression is: \(x^{2/3} - 4x^{5/6} + 8x^{1/3} - 4x^{1/2} + 7\).

Step 1: Use the Distributive Property

To multiply \((\sqrt[3]{x}+1)(\sqrt[3]{x}-4 \sqrt{x}+7)\), use the distributive property, also known as the FOIL method for binomials. Distribute each term in the first binomial across each term in the second binomial. This gives you: \[\sqrt[3]{x}(\sqrt[3]{x} - 4 \sqrt{x} + 7) + 1(\sqrt[3]{x} - 4 \sqrt{x} + 7).\]

Step 2: Distribute Each Term

Distribute the terms inside: - \( \sqrt[3]{x} \cdot \sqrt[3]{x} = x^{2/3}\) - \( \sqrt[3]{x} \cdot (-4\sqrt{x}) = -4x^{5/6}\)- \( \sqrt[3]{x} \cdot 7 = 7x^{1/3}\)Now add the contributions from \(+1\):- \( 1 \cdot \sqrt[3]{x} = x^{1/3}\)- \( 1 \cdot (-4\sqrt{x}) = -4x^{1/2}\)- \( 1 \cdot 7 = 7\)

Step 3: Combine Like Terms

Combine like terms by looking at the powers of \(x\):- The terms with \(x^{2/3}\) and \(x^{1/3}\) do not have similar terms, so they remain as they are.- The term \(-4x^{5/6}\) remains alone.- Combine \(7x^{1/3}\) and \(x^{1/3}\) to get \(8x^{1/3}\).- There is only one constant term: \(7\). The simplified expression is: \[x^{2/3} - 4x^{5/6} + 8x^{1/3} - 4x^{1/2} + 7\]

Key Concepts

Distributive Property

Multiplying polynomials involves using a foundational algebraic concept called the distributive property. Simply put, the distributive property is a rule that tells you how to handle multiplication over addition or subtraction inside parentheses. In this context, you're multiplying two binomials, which means you use the distributive property to expand the expression.

When you apply this rule, each term in the first binomial must be multiplied by each term in the second binomial. This ensures that no potential products are missed, which is essential for arriving at the correct result.

In our example

• First, take \ \(\sqrt[3]{x}\ \) and multiply it with each term within the second set of parentheses, \ (-4\sqrt{x}+7)\.
• Do the same for the other term \ (+1)\, making sure every term in the second binomial is covered.

Applying the distributive property correctly is crucial as it lays the groundwork for the rest of the problem solving.

Exponents

After distributing the terms, the next challenge is dealing with exponents in the expression. Exponents are shorthand for repeated multiplication of the same number by itself, and they follow their own set of rules. These rules help combine terms efficiently and accurately.

For instance, when multiplying expressions like \( \sqrt[3]{x} \times \sqrt[3]{x} \), use the rule that states \(a^m \times a^n = a^{m+n}\). So in this case, you get \(x^{2/3}\). Similarly:

• \( \sqrt[3]{x} \cdot (-4\sqrt{x}) = -4x^{5/6} \)
• \( \sqrt[3]{x} \cdot 7 = 7x^{1/3} \)
• \( 1 \cdot \sqrt[3]{x} = x^{1/3} \)
• \( 1 \cdot (-4\sqrt{x}) = -4x^{1/2} \)
• \( 1 \cdot 7 = 7 \)

Adjusting exponents correctly during multiplication helps avoid errors and keeps the expression organized.

Simplification of Expressions

With distributed terms and adjusted exponents, the last step is to simplify the expression. Simplification involves combining like terms and ensuring the expression is as neat as possible.

"Like terms" are terms that contain the same variable raised to the same power. By combining them, you improve the clarity and conciseness of your final answer.

In our case:

• Start with \(x^{1/3}\). Combine \(+7x^{1/3}\) and \(x^{1/3}\) to make \(8x^{1/3}\).
• It's important to line up and add the coefficients of like terms, serving to tidy up the polynomial.
• The final, simplified form of this particular expression is \[x^{2/3} - 4x^{5/6} + 8x^{1/3} - 4x^{1/2} + 7\}\].

Keeping things simplified makes it easier to read and understand, especially as equations get longer or more complex. Staying organized with terms helps avoid mistakes and misinterpretations.