Problem 18 - Algebra 1

Question

Evaluate the exponential expression. Write fractions in simplest form $$4\left(4^{-2}\right)$$

Step-by-Step Solution

Answer

The simplified result of the exponential expression \(4(4^{-2})\) is \(\frac{1}{4}\).

Step 1: Understand the negative exponent

Negative exponent means that instead of multiplying, the base will be divided. So \(4^{-2}\) can be interpreted as \(\frac{1}{4^2}\). It means 'one divided by the base to the power of 2'.

Step 2: Simplify the exponential expression

Solve \(\frac{1}{4^2}\). The base is 4 and the exponent is 2. Calculate 4 to the power of 2 which is 16. So the expression simplifies to \(\frac{1}{16}\).

Step 3: Multiply the result by 4

It is given that the result of \(4^{-2}\) is to be multiplied by 4. Multiplying \(\frac{1}{16}\) by 4 gives \(\frac{4}{16}\).

Step 4: Simplify the final result

The last step is to simplify the fraction to its simplest form (if applicable). The fraction we obtained, \(\frac{4}{16}\), simplifies to \(\frac{1}{4}\) by dividing both the numerator and the denominator by 4.

Key Concepts

Negative Exponent

When you encounter a negative exponent, it indicates an inverse operation to the basic principle of exponents. Instead of multiplying the base by itself for positive exponents, a negative exponent represents division. For example, the expression \(4^{-2}\) translates to \(\frac{1}{4^2}\), which essentially means 'one divided by the base raised to the power of the positive exponent'. Understand that any non-zero number with a negative exponent can be converted in this way, making the process of dealing with negative exponents a matter of flipping the fraction.

The fundamental principle here is that \(a^{-n} = \frac{1}{a^n}\), where \(a\) is the base and \(n\) is the positive equivalent of the negative exponent. This concept is pivotal not only in simplifying such expressions but also in understanding the broader concepts of inverses in mathematics.

Exponential Expression Evaluation

To evaluate an exponential expression, you need to follow certain rules and operations systematically. For instance, with \(4\left(4^{-2}\right)\), we start by solving the expression within the parentheses first, which is the negative exponent. After converting the negative exponent into a fraction, as mentioned earlier, we evaluate the expression \(4^2\) to obtain 16. Thus, \(4^{-2}\) simplifies to \(\frac{1}{16}\). Remember that when an exponent applies to a number, it means you multiply that number by itself as many times as the exponent indicates. The next steps involve carrying out any additional operations indicated, such as multiplication or division.

This methodical approach ensures that the exponential expressions are evaluated correctly. Having a solid understanding of exponents and the order of operations (PEMDAS/BODMAS) is key.

Simplifying Fractions

Simplifying fractions is a crucial process in mathematics that can make other operations easier. To simplify a fraction, you divide the numerator and the denominator by their greatest common divisor (GCD). This reduces the fraction to its simplest form. For example, in the expression \(\frac{4}{16}\), the GCD of 4 and 16 is 4. Dividing both the numerator and denominator by 4, we get \(\frac{1}{4}\).

The goal of simplifying fractions is to find the simplest way to express the same value. Simpler fractions are easier to understand, compare, and use in subsequent calculations. Mastery of simplifying fractions involves being familiar with basic multiplication and division, and understanding common factors and divisibility.

Mathematical Operations with Exponents

Dealing with mathematical operations that involve exponents can appear daunting, but it follows clear rules. When multiplying like bases, you add the exponents. Conversely, when dividing like bases, you subtract the exponents. It’s important to note that these rules only apply to expressions with the same base. For example, \(a^m \times a^n = a^{m+n}\) and \(\frac{a^m}{a^n} = a^{m-n}\), where \(a\) represents the base.

Other operations include raising a power to another power, in which case you multiply the exponents \((a^m)^n = a^{mn}\), and handling exponents when distributing over multiplication \(a^m \times b^m = (ab)^m\). These rules and properties form the essential toolkit for manipulating expressions with exponents and play a pivotal role in algebra and calculus.