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Chapter 9: Problem 10

Simplify the expression. $$6 \sqrt[3]{3}-5 \sqrt[3]{3}$$

Short Answer

Expert verified
\(\sqrt[3]{3}\)

Step by step solution

01

Identify Like Terms

The like terms in the given expression are \(6 \sqrt[3]{3}\) and \(-5 \sqrt[3]{3}\). Both terms are similar because they contain the same variable part, which is \(\sqrt[3]{3}\). We can perform operations (addition or subtraction) on these terms.
02

Subtract Coefficients

To simplify the expression, subtract the coefficient of the second term from that of the first term. This gives us \(6 - 5 = 1\).
03

Write Final Expression

Combine the result of the subtraction with the variable part. This gives us the simplified form of the original expression, which is \(1 \sqrt[3]{3}\) or simply \(\sqrt[3]{3}\).

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Key Concepts

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

Like Terms
When we talk about like terms, we're referring to parts of an expression that have the same variable and the same exponent. In our exercise, we have two terms: \(6 \sqrt[3]{3}\) and \(-5 \sqrt[3]{3}\). What makes them like terms is the \(\sqrt[3]{3}\) part, which is identical in both terms. This part is known as the 'variable part'.
  • To identify like terms, look for terms that have the exact same radicals and root numbers (such as cube roots or square roots).
  • In polynomial expressions, like terms have the same variable raised to the same power.
Recognizing like terms is crucial because it allows us to simplify expressions. Instead of handling each term separately, like terms can be combined by performing arithmetic operations on their coefficients.
Coefficients
Coefficients are the numbers in front of the variables or expressions, and they determine how many times you are taking that variable or term. In the expression \(6 \sqrt[3]{3} - 5 \sqrt[3]{3}\), the coefficients are 6 and -5.
  • They are numerical factors that precede the radical or variable in an expression.
  • When simplifying like terms, you subtract or add their coefficients.
In our exercise, simplifying involves subtracting these coefficients: taking 6 and subtracting 5 from it, which gives us 1. Whatever the coefficients are, they dictate how we combine the like terms in an expression. The final result rests on solving for the combined coefficient with the remaining radical expression.
Cube Roots
Cube roots are an important mathematical concept used to find a number that, when multiplied by itself three times, equals the original number under the root. In the expression given in the exercise, \(\sqrt[3]{3}\) signifies the cube root of 3.
  • Cube roots provide an easy way to express parts of a number that can be multiplied thrice to reach a specified value.
  • This expression tells us to find a number that, when cubed (raised to the power of 3), will return the value 3.
Understanding cube roots helps us simplify expressions in algebra. They often appear in equations that require finding a side of a perfect cube or simplify expressions involving radicals. Hence, recognizing cube roots enables smoother simplification and calculation processes in tasks involving radical expressions.

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