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Chapter 5: Problem 15

Simplify each numerical expression. \(2^{7} \cdot 2^{-3}\)

Short Answer

Expert verified
The simplified expression is 16.

Step by step solution

01

Identify Like Bases

Notice that both terms in the expression \(2^{7} \cdot 2^{-3}\) have the same base, which is \(2\). This allows us to apply the laws of exponents.
02

Apply the Product of Powers Rule

Use the product of powers rule which states \(a^m \cdot a^n = a^{m+n}\). Here, \(a = 2\), \(m = 7\), and \(n = -3\). So, \(2^{7} \cdot 2^{-3} = 2^{7 + (-3)} = 2^{4}\).
03

Simplify the Exponent

Now simplify the exponent: \(2^{7-3} = 2^{4}\). The exponent simplifies to 4.
04

Evaluate the Expression

Calculate \(2^{4}\), which is \(2 \times 2 \times 2 \times 2 = 16\).

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Key Concepts

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

Product of Powers Rule
When working with exponents, understanding the Product of Powers Rule can simplify complex expressions. This rule is especially handy when you encounter expressions with the same base. In simple terms, if you have the same base being multiplied, you can add the exponents together.
Imagine you have an expression like \(a^m \cdot a^n\). According to the rule, you can rewrite it as \(a^{m+n}\). This applies to any numbers as long as the base number is identical. For example, in \(2^7 \cdot 2^{-3}\), the base is 2. Thus, by applying the Product of Powers Rule, we add 7 and -3.
  • It simplifies calculations.
  • Helps in reducing complex problems to simpler ones.
  • Essential for algebra and higher mathematics.
Numerical Expressions
Numerical expressions involve numbers and operations like addition, subtraction, or exponentiation. In these expressions, you'll often look to simplify or evaluate them to find a single number. For instance, in the expression \(2^7 \cdot 2^{-3}\), you encounter exponents, which involve repeated multiplication.
To handle such expressions, recognize unique components like terms and exponents. Our task is to follow specific rules of arithmetic to simplify the expression effectively. This will often consist of working with numbers and arithmetic operations.
  • Identify base numbers and exponents.
  • Apply rules systematically to ensure accuracy.
  • Evaluate step by step to maintain clarity.
Simplifying Expressions
Simplifying expressions allows you to condense them into a more manageable form. The goal here is to reduce the expression as much as possible while maintaining its integrity.
In our case, we started with \(2^7 \cdot 2^{-3}\), which under the Product of Powers Rule becomes \(2^{4}\). This further simplifies to the numerical result of 16.
Here's how we proceed logically:
  • Combine like terms or exponents using established rules.
  • Transition from abstract exponents to concrete numbers when possible.
  • Compute the simplified expression into its final numerical form for clarity.

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Most popular questions from this chapter

For Problems \(19-32\), write each of the following in ordinary decimal notation. $$ (3.14)\left(10^{10}\right) $$

Convert each number to scientific notation and perform the indicated operations. Express the result in ordinary decimal notation. $$ \frac{66,000,000,000}{0.022} $$

For Problems \(1-18\), write each of the following in scientific notation. \(9126.21\)

Perform the indicated operations and express answers in simplest radical form. $$ \sqrt{2} \sqrt[4]{2} $$

For Problems \(1-18\), write each of the following in scientific notation. For example, $$ 27,800=(2.78)\left(10^{4}\right) $$ 89
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