Problem 36 - College Algebra

Question

Simplify each exponential expression $$ \frac{x^{30}}{x^{10}} $$

Step-by-Step Solution

Answer

The simplified form of the given exponential expression \(\frac{x^{30}}{x^{10}}\) is \(x^{20}\).

Step 1 Identify and Apply Exponential Rules

We know that \(\frac{x^{m}}{x^{n}} = x^{m-n}\) by the rule of exponents. We will apply this rule to the given expression \(\frac{x^{30}}{x^{10}}\).

Step 2 Simplify the Expression

By the rule applied, we get \(x^{30-10}\).

Step 3 Final Simplification

Upon subtracting the exponents, our final answer is \(x^{20}\).

Key Concepts

Exponential Rules

Exponential rules are essential tools in algebra. They help us simplify expressions by using properties of exponents. One fundamental rule is the quotient rule: \( \frac{x^m}{x^n} = x^{m-n} \). This rule is used whenever you divide two expressions with the same base. It allows you to subtract the powers, simplifying the entire expression into a form that’s easier to understand or compute. There are other important rules too:

• Product rule: \( x^m \times x^n = x^{m+n} \), which states you add the exponents when multiplying bases that are the same.
• Power rule: \((x^m)^n = x^{m \times n}\), where you multiply the exponents when taking a power to another power.
• Zero exponent rule: \( x^0 = 1 \), which tells us any base (except zero) raised to the power of zero is 1.

Understanding and applying these rules helps to streamline the process of working with algebraic expressions.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form. It makes math problems easier to solve and understand. The process can include condensing terms, applying algebraic operations, and using exponential rules.In our example: \( \frac{x^{30}}{x^{10}} \), we used the quotient rule to subtract the exponent in the denominator from the exponent in the numerator. This step-by-step application of rules transforms a seemingly complex expression into \( x^{20} \), a simpler and more manageable form.To simplify expressions effectively:

• Identify like terms or parts of expressions that can be combined.
• Apply correct exponential and algebraic rules.
• Double-check each step to ensure accuracy.

Simplifying expressions not only makes solving equations easier but also helps in understanding underlying mathematical concepts better.

Algebraic Expressions

Algebraic expressions form the foundation of algebra and consist of variables, numbers, and operations. They can be simple, like \( x + 2 \), or complex, like \( \frac{5x^2 - 3x + 4}{2x - 1} \). The ability to manipulate and simplify these expressions is crucial in both solving equations and understanding mathematical relationships.Key elements of algebraic expressions:

• Variables: Symbols such as \( x, y, \) or \( z \) that represent unknown values or quantities.
• Coefficients: Numbers that are multiplied by variables. In \( 3x \), 3 is the coefficient.
• Constants: Numbers on their own, without variables. In \( x + 5 \), 5 is a constant.
• Operators: Mathematical symbols that denote operations like addition (+), subtraction (-), multiplication (\( \times \)), and division (\( \div \)).

Mastering the manipulation of algebraic expressions, such as applying exponential rules and simplifying, allows you to solve equations, model real-world scenarios, and develop a deeper understanding of algebra itself.