Problem 73 - Algebra 2

Question

Simplify each expression. 32\(\cdot 256^{-\frac{1}{2}}\)

Step-by-Step Solution

Answer

\nThe simplified expression is 2.

Step 1: Understanding the Negative Exponent

Negative exponent means we have to take the reciprocal of the base. Therefore, \(256^{-1/2}\) essentially means 1 over the square root of 256.

Step 2: Calculating the Square Root

The square root of 256 is 16. So, \(256^{-1/2}\) simplifies to 1/16.

Step 3: Final Simplification

Multiply 32 by 1/16. This simplifies to 2, which is the final answer.

Key Concepts

Understanding Negative Exponents

When we encounter a negative exponent, it might initially seem a bit confusing. Simply put, a negative exponent indicates that we need to take the reciprocal of the base raised to the positive of that exponent.

• For a general number, let's say \(a\), with an exponent \(-n\), the expression \(a^{-n}\) can be transformed into \(\frac{1}{a^n}\).
• This rule helps us transform what initially looks complex into something more manageable.
• In the context of our example, \(256^{-1/2}\) becomes \(\frac{1}{256^{1/2}}\), simplifying the understanding of the problem.

In this way, negative exponents help us work with more friendly fractions, instead of unwieldy decimal numbers.

Calculating Square Roots

Square roots are another important concept in algebra simplification. The square root of a number is a value that, when multiplied by itself, gives the original number.

• For our example, we need to find the square root of 256. So, you look for a number that, when squared, results in 256.
• The number 16 multiplied by itself (16 × 16) equals 256. Therefore, \(\sqrt{256} = 16\).
• By converting \(256^{-1/2}\) into \(\frac{1}{\sqrt{256}}\), it becomes \(\frac{1}{16}\) once the square root is calculated.

Understanding square roots allows us to simplify expressions and easily handle fractional exponents like the one in our given solution.

Finding Reciprocals

Reciprocals are a basic yet crucial aspect of working with fractions and simplifying expressions, especially when negative exponents come into play.

• The reciprocal of a number is simply 1 divided by that number. For example, the reciprocal of 16 is \(\frac{1}{16}\).
• This concept is especially important when dealing with expressions with negative exponents or fractions.
• By understanding reciprocals, you can easily rewrite expressions like \(256^{-1/2}\) as \(\frac{1}{16}\).

In the final step of simplification, we used this concept by multiplying the reciprocal \(\frac{1}{16}\) by 32 to get the final simplified answer.