Problem 29 - Algebra 1: Concepts and Skills

Question

Copy and complete the statement. \(\left(x^{3}\right)^{3}=x^{?}\)

Step-by-Step Solution

Answer

The missing exponent is 9.

Step 1: Identify the base and exponents

Here, the base is \(x\), the inner exponent is 3, and the outer exponent is also 3.

Step 2: Apply the power of a power rule

According to the rule, we multiply the inner exponent with the outer exponent: \(3*3 = 9\).

Step 3: Write the complete expression

So, \(\left(x^{3}\right)^{3} = x^{9}\). Therefore, the question mark should be replaced with 9.

Key Concepts

Exponents

Exponents are a crucial part of algebraic expressions and play a significant role in mathematical operations. An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, if we have \(x^3\), it means \(x\) is multiplied three times: \(x \times x \times x\). Exponents make it easier to write repeated multiplications in a compact form.

There are several rules related to exponents that we often use:

• Power of a power: This rule states that when you raise an exponential expression to another power, you multiply the exponents.
• Product of powers: When multiplying two expressions with the same base, you add the exponents.
• Quotient of powers: This rule states that when dividing expressions with the same base, you subtract the exponents.

Understanding these rules helps simplify complex algebraic expressions involving exponents.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators like addition, subtraction, multiplication, and division. They form the language of algebra and are used to represent real-world problems in a form that can be solved or analyzed algebraically.

In the context of this exercise,

• \(x\) represents a variable. It's a placeholder for any value.
• The expression \(x^3\) suggests that \(x\) will be multiplied by itself three times.

Variables often work with coefficients in expressions. Coefficients are numbers placed in front of variables to scale them up or down. For example, in the expression \(3x^3\), 3 is the coefficient of \(x^3\).

Algebraic expressions can be manipulated using different rules, including the power of a power rule, which helps simplify expressions like \((x^3)^3\). Mastering these concepts allows you to solve equations and inequalities effectively.

Mathematical Operations

Mathematical operations are the basic procedures like addition, subtraction, multiplication, and division. They form the foundation of most mathematical problem-solving. In algebra, these operations also extend to include working with exponents and variables.

When dealing with exponents in mathematical operations, several rules are essential:

• Multiplying powers: Multiply the coefficients and add the exponents if the bases are the same.
• Dividing powers: Divide the coefficients and subtract the exponents when bases match.
• Power of a power: Multiply the exponents.

Knowing how to effectively apply these operations is key to simplifying and solving algebraic expressions. For example, applying the power of a power rule correctly in the exercise converts \((x^3)^3\) into \(x^9\).

Understanding these mathematical operations ensures you handle algebraic expressions confidently, whether you're multiplying complex terms or simplifying long equations.