Problem 69 - Algebra 1: Concepts and Skills

Question

Evaluate the expression \(\left(4^{-1}\right)^{-2}\) F. \(\frac{1}{64}\) G. \(\frac{1}{16}\) H. 16 J. 64

Step-by-Step Solution

Answer

H. 16

Step 1: Simplify the Inner Exponent

Start by simplify the inner part of the operation \(\left(4^{-1}\right)\). The negative exponent means that the base is flipped from numerator to denominator: \(\left(4^{-1}\right) = \frac{1}{4}\)

Step 2: Applying the Outer Exponent

Then apply the outer exponent \(-2\) to the value obtained in Step 1. \((-2)\) as an exponent acts like a normal 2 exponent but flips the fraction: \(\left(\frac{1}{4}\right)^{-2} = 4^{2}\)

Step 3: Calculate Result

Finally, calculate \(4^{2}\) to get: \(4^{2} = 16\)

Key Concepts

Negative Exponents

Negative exponents might seem tricky at first, but they actually represent a simple concept. When you see a negative exponent, it indicates the reciprocal of the base raised to the positive of that exponent. For example, when you have a number like \(a^{-n}\), it translates to \(\frac{1}{a^n}\). This means you flip the base from the numerator to the denominator and then apply the positive exponent.

Here's a quick way to think about it: a negative exponent tells you to "take the reciprocal and then apply the positive exponent." It's just like saying "flip it, and exponentiate it." This fundamental principle is at work whenever you see negative exponents in expressions, and understanding this can help simplify many algebraic problems.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This includes operations like combining like terms, using the distributive property, and especially in cases like this, properly dealing with exponents. Let's use the original exercise as an example:

First, you simplify inside the parentheses by addressing any negative exponents. For \(4^{-1}\), you recognize it means "one over four," or \(\frac{1}{4}\). Once simplified, move on to other operations like applying outer exponents. Continuing with the example, \(\left(\frac{1}{4}\right)^{-2}\) needs you to flip the fraction due to another negative exponent, resulting in \(4^{2}\).

Simplifying expressions not only makes them shorter and easier to work with but also helps in better understanding the underlying mathematics. Breaking down the process step-by-step ensures accuracy in your final result.

Algebra Concepts

Algebra is the foundation of many areas in mathematics and incorporates numerous key concepts. Understanding algebra revolves around manipulating symbols and numbers to solve problems or represent real-world scenarios. Key components of algebra include working with variables, expressions, equations, and different operations such as addition, subtraction, multiplication, and division.

In the context of exponents, algebra helps in organizing operations and applying rules of exponents systematically. For example, knowing how to simplify an expression with negative exponents or calculate a positive or negative power is crucial. Algebra provides a systematic approach to solve such equations by applying known rules effectively.

• Use properties of operations to simplify expressions.
• Apply the rules of exponents for straightforward solutions.
• Set up equations based on patterns or given data.

These aspects of algebra make it not only essential for mathematics but also highly applicable in various scientific and engineering fields.