Question

Evaluate the expression. Write fractional answers in simplest form.\(8 \cdot 2^{-2} \cdot 4^{-1}\)

Short answer

The simplified value of the given expression is 1/2 or \(0.5\) in decimal form.

Step-by-step solution

Describe the Approach

To evaluate the given expression, start by simplifying each term and respect the order of operations. Any number with a negative exponent can be written as the reciprocal of the same number with a positive exponent. Thus, \(2^{-2}\) can be written as \(1/2^{2}\) and \(4^{-1}\) can be written as \(1/4^{1}\). Then, perform the multiplication.

Simplify Each Term

First, simplify \(2^{-2}\) and \(4^{-1}\) by rewriting them as reciprocals.\n\n So, \(2^{-2} = 1/2^{2} = 1/4\) and \(4^{-1} = 1/4^{1} = 1/4\).\n\nThe expression now looks like this: \(8 \cdot 1/4 \cdot 1/4\).

Perform the Multiplication

Multiply from left to right: \(8 \cdot 1/4 = 2\). Then, multiply the result by \(1/4\) to get \(2 \cdot 1/4 = 0.5\) or \(1/2\) when expressed as a fraction.

Key concepts

Order of Operations

Understanding the order of operations is essential when simplifying mathematical expressions, especially those involving exponents. This rule, often remembered using the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), guides you through the steps needed to evaluate a complex expression.

Let's apply this rule to the expression given in our exercise: \(8 \times 2^{-2} \times 4^{-1}\). According to the order of operations, we address the exponents before the multiplication. The negative exponents indicate we should find the reciprocal of the base raised to the positive exponent (as we'll further discuss in the 'Reciprocal of Exponents' section). After rewriting the terms with negative exponents as reciprocals, we multiply the terms from left to right. If we had additional operations, like addition or subtraction, they would be handled after all multiplication and division are complete.

Reciprocal of Exponents

Negative exponents can sometimes be tricky, but once you understand the concept of reciprocals, they become much more manageable. A negative exponent, like \(-n\), is telling us to take the reciprocal, or the 'flip', of the base raised to the positive exponent \(n\). In short, \(b^{-n} = \frac{1}{b^n}\).

For instance, in the given exercise, \(2^{-2}\) becomes \(\frac{1}{2^{2}}\), which simplifies to \(\frac{1}{4}\). Likewise, \(4^{-1}\) becomes \(\frac{1}{4^1}\) or simply \(\frac{1}{4}\). By transforming negative exponents into their reciprocal form, we can proceed to multiply these values along with other terms in the expression as if they were positive exponents or whole numbers.

Fractional Notation

Using fractional notation means expressing numbers not as decimals, but as fractions—ratios of whole numbers. This format is often preferred for exactness and is very common in mathematical, scientific, and engineering calculations. When simplifying expressions, final answers may sometimes be more appropriate or insightful if left as fractions instead of being converted to decimal form.

In the context of our example, the arithmetic operations lead us to an answer of 0.5, which we can express in fractional notation as \(\frac{1}{2}\). This form is simpler and more informative, especially when dealing with ratios or proportions. It's also important when simplifying fractions that you express them in their simplest form, meaning that the numerator and denominator have no common divisors other than 1.