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Chapter 1: Problem 13

Simplify. $$ 0.5^{\wedge} 2 $$

Short Answer

Expert verified
0.5^2 = 0.25

Step by step solution

01

Understand the Problem

We need to simplify the expression \(0.5^{\wedge}2\), which can also be written in mathematical notation as \(0.5^2\). This is an exponential expression where 0.5 is the base and 2 is the exponent.
02

Apply the Power Rule

To simplify \(0.5^2\), we use the rule of exponents which states that \(a^b\) means multiplying the base \(a\) by itself \(b\) times. So, \(0.5^2 = 0.5 \times 0.5\).
03

Perform the Multiplication

Now, calculate \(0.5 \times 0.5 = 0.25\). This is the result of multiplying the base 0.5 by itself once, as indicated by the exponent 2.

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Key Concepts

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

Simplifying expressions
Simplifying expressions involves rewriting them in a more straightforward or more easily manageable form. When you have expressions with exponents, like \(0.5^2\), you need to break them down into their basic components. This process combines mathematical rules such as the power rule and arithmetic operations like multiplication.
  • Identify the expression: In our case, we start with \(0.5^2\).
  • Reinterpret the expression using exponent rules.
  • Perform any necessary arithmetic operations to find the simplified result.
You'll find simplifying expressions useful across mathematics, especially in algebra, where you often combine like terms or factor numbers to make equations easier to solve.
Power rule
The power rule is a fundamental tool in dealing with exponents. It states that raising a number to a power means multiplying that number by itself as many times as the exponent indicates. Here, in \(0.5^2\), the base is 0.5, and the exponent is 2. This means you multiply 0.5 by itself once. Understanding this rule helps you put exponential expressions into simpler forms:
  • The expression \(a^b\) means that base \(a\) is used as a factor \(b\) times.
  • In our example: \(0.5^2 = 0.5 \times 0.5\).
The power rule is a key piece of knowledge necessary for figuring out exponential growth models, scientific calculations, and more.
Multiplication of decimals
When we multiply decimals, like in our example \(0.5 \times 0.5\), it's helpful to think about it as multiplying whole numbers. First, ignore the decimal points and multiply the numbers as if they were whole numbers. In this case:
  • Multiply 5 by 5 to get 25.
Next, count the total number of digits to the right of the decimal points in the original numbers. Here, each 0.5 has one digit after the decimal point, so we count two decimals in total. This tells us the product must also have two digits to the right of the decimal.Finally, apply the decimal places to the product:
  • Thus, the result of \(5 \times 5 = 25\) becomes 0.25 when considering the decimal places.
Multiplying decimals accurately is an essential skill, especially in scientific measurements, finance, and everyday math activities.

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