Problem 39 - Algebra 2

Question

Write each expression in simplest form. Assume that all variables are positive. $$\left(x^{-4}\right)^{7}$$

Step-by-Step Solution

Answer

The simplest form of the given expression is \(1/x^{28}\)

Step 1: Identify the Property of Exponents to be Applied

The given expression is \( \left(x^{-4}\right)^{7} \). This is a case of 'power of a power', meaning an exponentiated quantity itself raised to a power. In such cases, we multiply the exponents.

Step 2: Apply the Property of Exponents

Let's apply the rule in this case: \(-4*7 = -28\). So, the expression reduces to \(x^{-28}\).

Step 3: Transform Negative Exponent into Positive

Negative exponent indicates reciprocation. So, \(x^{-28}\) is same as \(1/x^{28}\). This is the simplest form of the given expression.

Key Concepts

Properties of Exponents

Exponents can sometimes seem tricky, but they follow a few simple properties that make working with them easier. Understanding these properties helps simplify equations and expressions involving powers. Here are some of the most commonly used properties:

• Product of Powers: When you multiply two expressions with the same base, you add the exponents. For example, if you have \(a^m \times a^n\), it simplifies to \(a^{m+n}\).
• Power of a Power: This one is crucial for our original exercise. When you have a power raised to another power, you multiply the exponents. For example, \((a^m)^n\) becomes \(a^{m \times n}\).
• Power of a Product: When each factor in a product is raised to an exponent, you can distribute the exponent over the factors. For example, \((ab)^n\) becomes \(a^n\times b^n\).
• Quotient of Powers: When dividing two expressions with the same base, subtract the exponents. For example, \(a^m / a^n\) simplifies to \(a^{m-n}\).

By understanding these properties, you'll have a strong foundation for tackling more complex problems involving exponents.

Negative Exponents

Negative exponents might look intimidating, but they actually make life easier by turning expressions into fractions. Let's break down what a negative exponent does.
A negative exponent indicates the reciprocal of the base raised to the positive exponent. For instance, \(x^{-n}\) is equivalent to \(1/x^n\). This is a crucial concept when simplifying expressions.
Moving the base with a negative exponent to the other side of the fraction line (from numerator to denominator or vice versa) changes the sign of the exponent. This means that \(a^{-3}\) becomes \(1/a^3\) and vice versa, \(1/a^{-3}\) becomes \(a^3\).
Using negative exponents helps convert large or unwieldy numbers into easier, more manageable forms. Always remember: the negative sign in an exponent indicates an inverse operation.

Algebraic Expressions

Algebraic expressions are combinations of variables and numbers, connected by operations like addition, subtraction, multiplication, and division.
When dealing with exponents, there are special rules that help simplify these complex-looking expressions. Using these rules makes it easier to solve and manipulate them.
In the expression \(\left(x^{-4}\right)^{7}\), for instance, we're working with variables raised to powers. The expression simplifies to \(x^{-28}\), which further simplifies to \(1/x^{28}\) by applying our understanding of exponents.
Remember to always apply the appropriate properties of exponents and transform them as needed (like converting negative exponents) to simplify algebraic expressions effectively.

Power of a Power

When you encounter the power of a power property in exponents, it means you're dealing with a situation where you have an expression in the form \((a^m)^n\).
According to the power of a power rule, you multiply the exponents. This simplifies the expression to \(a^{m\times n}\).
This rule is especially helpful because it reduces complex expressions into simpler, easier-to-work-with forms. It's the key concept we used to solve the original exercise where \((x^{-4})^7\) was simplified to \(x^{-28}\) by multiplying the exponents \(-4\) and \(7\).
Always remember this property when dealing with nested exponents; it makes simplification straightforward and efficient. Practice makes perfect, so keep applying this rule to ensure you have a solid grasp on the concept.