Problem 28 - Algebra and Trigonometry

Question

Simplify each exponential expression. $$ x^{-5} \cdot x^{10} $$

Step-by-Step Solution

Answer

The simplified form of the given exponential expression is \(x^5\).

Step 1: Identify Exponents and Base

The terms have the same base, which is \(x\), and their exponents are -5 and 10. They are multiplied together, so we use the rule of exponents that allows us to add the exponents when multiplying terms with the same base.

Step 2: Apply Rule of Exponents

We add the exponents together: -5 + 10 = 5.

Step 3: Write Final Simplified Expression

The multiplication simplifies to a single term, \(x^5\).

Key Concepts

Rule of Exponents

When working with exponential expressions, understanding the rule of exponents is crucial for simplification. This rule is particularly useful when you have the same base being raised to different powers. One fundamental property says that when multiplying such expressions, you should add the exponents. In an algebraic form, if you have a base 'a' raised to an exponent 'm' and multiplied by the same base 'a' raised to an exponent 'n', the simplified form will be 'a' raised to the exponent 'm+n'.

For example, the expression \(a^m \cdot a^n\) becomes \(a^{m+n}\). This is exactly what we apply in the exercise with the base 'x' to combine \(x^{-5} \cdot x^{10}\) into a single term. Remember, this rule applies only when the base is the same and the terms are being multiplied. It’s a powerful tool that streamlines many algebraic processes and makes handling exponential terms much easier.

Exponent Addition

Exponent addition comes into play when we're multiplying exponents with the same base; it's a direct application of the rule of exponents. To put it simply, when you want to multiply two exponential terms with the same base, instead of multiplying the bases, you keep the base the same and add the exponents. This shortcut saves you from working out the long-form multiplication and allows for quick simplification of expressions.

For instance, the exercise provided requires us to combine \(x^{-5}\) and \(x^{10}\). Following our rule, we add the exponents: \(-5 + 10 = 5\). Thus, the expression simplifies to \(x^5\). This shows how adding exponents can condense the expression and make it more digestible. It's an essential skill for students to master, making algebra much more approachable.

Algebraic Simplification

Algebraic simplification is the process of rewriting an expression in its simplest form. This makes it easier to understand or further manipulate mathematically. The steps usually involve applying algebraic rules like the rule of exponents, distributing multiplication across addition and subtraction, combining like terms, and factoring, where appropriate.

In our exercise, simplification was achieved by using the rule of exponents for exponent addition. No like terms or distribution was needed, showing that sometimes simplification can be a straightforward application of a single rule. Algebraic simplification is about knowing which rules to apply and when, helping to break down complex expressions into more manageable pieces.