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Chapter 0: Problem 31

Evaluate each exponential expression. $$\left(x^{3}\right)^{7}$$

Short Answer

Expert verified
\(\left(x^{3}\right)^{7} = x^{21}\)

Step by step solution

01

Applying the power of a power rule

The rule of exponents states that \((a^m)^n = a^{mn}\). Apply this rule to the given expression. Here, \(a = x\), \(m = 3\), and \(n = 7\). So, \(\left(x^{3}\right)^{7} = x^{3*7}\).
02

Simplify the expression

Simplify \(x^{3*7}\) by multiplying 3 and 7, which gives 21. So, \(x^{3*7} = x^{21}\).

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Key Concepts

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

Exponential Expression
An exponential expression is written with a base and an exponent, showcasing how many times the base is multiplied by itself. For example, in the expression \(x^3\), \(x\) is the base, and the exponent is 3, indicating that \(x\) is used as a factor three times: \(x*x*x\). These expressions depict exponential growth or decay, commonly seen in areas like compound interest, population growth, and radioactive decay. In the given exercise, we evaluate the exponential expression \( \left(x^{3}\right)^{7} \).
Rules of Exponents
Understanding the rules of exponents is crucial for simplifying complex expressions. These rules make dealing with exponential expressions more manageable. A critical rule is the power of a power rule, which states that when you raise a power to another power, you multiply the exponents. In mathematical form, \( (a^m)^n = a^{mn} \). Other rules include product of powers (\( a^m \cdot a^n = a^{m+n} \)) and quotient of powers (\( \frac{a^m}{a^n} = a^{m-n} \)), among others. The exercise applies the power of a power rule specifically.
Simplify Expression
To simplify an expression means to reduce it to its simplest form, making it easier to understand or solve. When simplifying expressions, especially those with exponents, we apply the relevant rules of exponents. For instance, simplifying \( \left(x^{3}\right)^{7} \) involves multiplying the exponents (power of a power rule) to find that \(x^{3}\cdot x^{7} = x^{21}\). Simplification helps in solving equations, simplifying calculations, and understanding relationships in algebraic expressions.
Mathematical Notation
Mathematical notation is a system of symbols used to express mathematical ideas. It serves as a concise language for mathematicians to communicate complex concepts efficiently. For example, exponents are written as a smaller number above and to the right of the base number, indicating the power to which the base is raised. The correct use of notation, such as \(x^{21}\) instead of writing \(x\) multiplied by itself 21 times, is fundamental for clarity and precision in mathematical communication. Learning to interpret and use this notation correctly is essential for every math student.

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