Problem 33 - Algebra and Trigonometry

Question

Simplify each exponential expression. $$ \left(x^{-5}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified form of the given exponential expression is \(x^{-15}\).

Step 1: Identify the power of a power property

Here, the power of a power property of exponents applies. This is because we have a term \((x^{-5})^3\). Let's apply this rule, which states that when an exponent is raised to another exponent, we multiply them. Thus, the operation becomes \(x^{-5 \cdot 3}\).

Step 2: Simplification

Now, we do the multiplication in the exponent. Doing the multiplication gives \(x^{(-5) \cdot 3} = x^{-15}\).

Key Concepts

Understanding the Power of a Power Property in Exponents

When studying algebra, you might come across terms raised to an exponent, which are then further raised to another exponent. To simplify such expressions, you can use the 'power of a power' property. This rule is remarkably straightforward: just multiply the exponents. For instance, if you have an expression like \( (a^m)^n \), where \( a \) is the base and \( m \) and \( n \) are the exponents, you apply it by multiplying \( m \) and \( n \) together to get \( a^{m \times n} \).

This rule not only makes calculations easier but also helps you understand how powers and exponents are related on a deeper level. To illustrate the rule using the exercise (\

Exponents and Their Properties

Exponents play a pivotal role in algebra, providing a compact way to represent repeated multiplication. An exponent, which is usually written as a small number above and to the right of a base number, tells you how many times to multiply the base by itself. For example, \( 4^3 \) means you multiply 4 by itself three times: \( 4 \times 4 \times 4 \).

Understanding exponents involves familiarizing yourself with several properties and rules. Negative exponents, like in the term \( x^{-5} \), indicate division instead of multiplication. In the exercise example, \( x^{-5} \) tells us that we are dealing with \( 1/(x^5) \), which means \( x \) is being divided by itself five times. The concept of 'zero exponent', where any number raised to the power of zero equals one, is another important property to know, along with the power of a power property discussed earlier.

Mastering Algebraic Simplification

Simplifying algebraic expressions is a fundamental skill in mathematics that involves reducing expressions to their most basic form without changing their value. Algebraic simplification often involves combining like terms, using the distributive property, and applying the rules of exponents, as we did in the given exercise.

Working with exponential expressions can at first seem daunting due to the various rules that apply. But by breaking it down step by step, as in the example of \( (x^{-5})^3 \), and by applying the power of a power property effectively, you learn to recognize patterns and start simplifying expressions with confidence. Remember to take the exercise step by step—first by applying the power of a power property and then by carrying out the multiplication of the exponents—to reach the simplified form \( x^{-15} \).