Problem 62 - Elementary and Intermediate Algebra

Question

Simplify by combining like radicals. $$ 12+\sqrt[3]{80}-\sqrt[3]{10,000}+4 $$

Step-by-Step Solution

Answer

The expression simplifies to \(16 - 8 \times \sqrt[3]{10}\).

Step 1: Combine Constants

First, identify and add up all the constants in the expression. Here, the constants are 12 and 4. So, adding them together gives us 16.

Step 2: Simplify Cube Roots

Now, simplify each cube root separately. Begin by factoring the numbers under the radical symbols. For \(\sqrt[3]{80}\), we find that \(80 = 2^4 \times 5\). Rewriting, we get \(\sqrt[3]{80} = \sqrt[3]{2^4 \times 5} = 2 \times \sqrt[3]{10}\). Next, for \(\sqrt[3]{10,000}\), we rewrite 10,000 as \(10^4\), so \(\sqrt[3]{10,000} = \sqrt[3]{10^4} = 10 \times \sqrt[3]{10}\).

Step 3: Combine Like Radicals

Now that we have the simplified form of the cube roots, \(2 \times \sqrt[3]{10}\) and \(10 \times \sqrt[3]{10}\), we can combine them as like terms. Combining gives us \((2 - 10) \times \sqrt[3]{10} = -8 \times \sqrt[3]{10}\).

Step 4: Write Final Simplified Expression

Finally, rewrite the expression by combining the constant terms and the simplified like radicals. The constants add up to 16, and the radicals simplify to \(-8 \times \sqrt[3]{10}\). Therefore, the expression simplifies to: \(16 - 8 \times \sqrt[3]{10}\).

Key Concepts

Combining Like Terms

When you come across terms in an expression that look similar, it's an invitation to combine them. These similar-looking terms are known as "like terms." In mathematics, like terms are variables or expressions that have exactly the same variable parts raised to the same power. For example, in the equation \(3x + 2x\), the terms \(3x\) and \(2x\) are like terms, because they both involve \(x\).

Combining like terms involves adding or subtracting the coefficients of these terms. In the original exercise, after simplifying the cube roots, we end up with terms \(2 \times \sqrt[3]{10}\) and \(10 \times \sqrt[3]{10}\), which are like terms. This is because they both involve the "cube root of 10." The operation we perform is subtraction of their coefficients \(2\) and \(10\), resulting in \(-8 \times \sqrt[3]{10}\).

This process helps to simplify expressions and is an essential skill in algebra for making calculations manageable and easier to understand.

Cube Roots

Cube roots are the inverse operation of cubing a number. When you "cube" a number, you raise it to the power of three. To "cube root" a number means finding a number that, when multiplied by itself three times, gives the original number. For instance, the cube root of 27 is 3, because \(3 \times 3 \times 3 = 27\).

In our exercise, we required simplifying expressions like \(\sqrt[3]{80}\) and \(\sqrt[3]{10,000}\). To simplify these, it's beneficial to factor the number under the cube root into primes or powers of numbers that are easily managed. For \(\sqrt[3]{80}\), we break it down to \(\sqrt[3]{2^4 \times 5}\), allowing us to simplify further by separating out a factor of \(2\).

• \(80 = 16 \times 5\)
• \(16 = 2^4\)

This process turns into \(2 \times \sqrt[3]{10}\). Similarly, simplifying \(\sqrt[3]{10,000}\) gives us \(10 \times \sqrt[3]{10}\), as \(10,000\) can be expressed as \(10^4\). Understanding and simplifying cube roots is crucial for working comfortably with radical expressions.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, of numbers or expressions. They can appear complex, but with practice, they become easier to manage. A radical expression is composed of a radical symbol \(\sqrt{}\) and a radicand, the number inside the symbol.

In algebra, simplifying radical expressions means removing factors that are "perfect" powers of the root outside of the radical. For example, taking the original radical \(\sqrt[3]{80}\), since this is a cube root, we look for cubes within the factorization of 80. After breaking down 80 into primes, we identify cubes like \(2^3\), which allows us to extract the \(2\) from the radical, simplifying \(\sqrt[3]{80}\) to \(2 \times \sqrt[3]{10}\).

• Factorizing helps identify parts that can be simplified.
• Using properties of exponents and roots can streamline complicated calculations.

Understanding the properties of radicals enhances our ability to simplify and manipulate expressions, paving the way for solving more complex algebraic equations successfully. Engaging with radical expressions equips students with vital mathematical skills useful across various branches of mathematics.