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Chapter 11: Problem 42

Evaluate each of the numerical expressions. $$2^{\frac{3}{2}} \cdot 2^{-\frac{1}{2}}$$

Short Answer

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Step by step solution

01

Apply the Product of Powers Property

To simplify the expression, use the product of powers property: \(a^m \cdot a^n = a^{m+n}\). Here, \(m = \frac{3}{2}\) and \(n = -\frac{1}{2}\). Calculate \(m + n\).
02

Simplify the Exponent

Add the exponents: \(\frac{3}{2} + \left(-\frac{1}{2}\right) = \frac{3}{2} - \frac{1}{2} = \frac{2}{2} = 1\).
03

Evaluate the Expression

Substitute the simplified exponent back into the expression: \(2^1\). This simplifies directly to \(2\).

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Key Concepts

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

Product of Powers
Understanding the 'product of powers' property helps you easily simplify expressions with the same base. This property states that when you multiply two exponential expressions with the same base, you simply add the exponents. For example, given two powers like \(a^m \cdot a^n\), you can rewrite them as \(a^{m+n}\).

Here's a quick breakdown:
  • Both expressions must have the same base, such as \(2^{\frac{3}{2}}\) and \(2^{-\frac{1}{2}}\) having the base 2.
  • Add the exponents: \(\frac{3}{2} + (-\frac{1}{2})\) simplifies to \(\frac{2}{2} = 1\).
  • Your new expression becomes \(2^1\).
By mastering this simple rule, you can handle more complex expressions easily!
Numerical Expressions
Numerical expressions involve numbers and operation symbols, such as addition, subtraction, multiplication, or division. When working with exponents, these expressions become a tad more interesting.

Let's consider the expression: \(2^{\frac{3}{2}} \cdot 2^{-\frac{1}{2}}\). Here, you are not just dealing with simple multiplication but with powers that add another layer of complexity.
  • The base here is 2, which repeats itself, making it easy to apply the product of powers.
  • Don't let fractional exponents intimidate you; focus on the operation, which in this case is adding the exponents due to the same base.
  • This will ease the path to simplify the expression as you will see in the next steps.
By practicing, you'll find that handling numerical expressions with exponents becomes second nature.
Simplifying Exponents
Simplifying exponents can seem daunting, but it often involves straightforward addition or subtraction, as illustrated here. In the example \(2^{\frac{3}{2}} \cdot 2^{-\frac{1}{2}}\), follow these steps:

First, use the product of powers property to join the exponents into a single term by adding them together: \(\frac{3}{2} + (-\frac{1}{2})\).

Doing this arithmetic gives you \(\frac{2}{2} = 1\).
  • Write the expression in its simplified form: \(2^1\).
  • Simplify further: any number to the power of 1 is itself, so this simplifies to 2.
This step-by-step method not only works in this context but also with larger or more complex powers. The key is always to address the exponents correctly. Focus on simple arithmetic principles, and you'll simplify exponents like a pro!

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Most popular questions from this chapter

Which method would you use to solve the equation \(x^{2}+4 x=-5\) ? Explain your reasons for making that choice.

For Problems \(13-34\), add or subtract the complex numbers as indicated. $$ (-9+7 i)-(-8-5 i) $$

For Problems \(35-54\), find each product and express it in the standard form of a complex number \((a+b i)\). $$ (4+2 i)(6+5 i) $$

For Problems \(13-34\), add or subtract the complex numbers as indicated. $$ (3+i)-(7+4 i) $$

Solve each of the following quadratic equations, and check your solutions. $$x^{2}+2 x+5=0$$
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