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Chapter 1: Problem 44

Simplify. \(16^{5 / 2}\)

Short Answer

Expert verified
1024

Step by step solution

01

- Understand the exponent

The expression has a fractional exponent \(\frac{5}{2}\). Recall that \(a^{m/n} = \sqrt[n]{a^m}\). So, for \(16^{5/2}\), you can rewrite the exponentiation as \( (16^{1/2})^5 \).
02

- Simplify the base

Rewrite \(16^{1/2}\) using the square root. Since the square root of 16 is 4, we have \(16^{1/2} = 4\). Therefore, the expression becomes \(4^5\).
03

- Compute the power

Calculate \(4^5\) by multiplying 4 by itself 5 times: \(4 \times 4 \times 4 \times 4 \times 4 = 1024\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Exponents
When you are faced with a fractional exponent, the first step is to break it down into more manageable parts. Fractional exponents can be tricky but remember the rule: \(a^{m/n} = \sqrt[n]{a^m}\). This means any base raised to a fractional exponent can be converted using roots and powers. In our example, \(16^{5/2}\) can be rewritten as \((16^{1/2})^5\).
Once you simplify a base using the square root or another root, it becomes much easier to handle. Practice this step to get comfortable transitioning from fractional exponents to simpler expressions. It'll make solving similar problems quick and intuitive!
Square Root
The square root is a key part of simplifying exponents, especially when the exponent is in the form \frac{m}{2}\. Recall that the square root of a number is simply a value that, when multiplied by itself, gives the original number. For example, the square root of \16\ is \4\ because \4 \times 4 = 16\.
Understanding how to work with square roots can make working with fractional exponents much simpler. In our problem, we rewrote \(16^{1/2}\) as \4\ because \(\sqrt{16} = 4\).
Once the base is simplified using the square root, you can easily proceed to the power calculation. This is especially useful for larger numbers or more complex expressions.
Power Calculation
Finally, you need to calculate the power once you have simplified the base. You've turned your problem into something more familiar! After rewriting \(16^{1/2}\) as \4\, the problem becomes \4^5\.
To calculate a power like \(4^5\), multiply \4\ by itself five times: \4 \times 4 \times 4 \times 4 \times 4 = 1024\. This step is straightforward if you break it down into smaller parts:
  • First, calculate \(4^2 = 16\).
  • Then, multiply by \4\ to get \(16 \times 4 = 64\).
  • Finally, multiply by \4\ again to reach \(64 \times 4 \times 4 = 1024\).
This logical sequence can make complex power calculations simple and manageable.

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Most popular questions from this chapter

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