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Chapter 5: Problem 71

Simplify each expression. $$ \frac{(4.5)^{2}}{100} $$

Short Answer

Expert verified
The expression simplifies to \(0.2025\).

Step by step solution

01

Expand the Exponent

Start by expanding the square of the numerator. Here, we have \[(4.5)^2 = 4.5 \times 4.5\] Calculate this product.
02

Calculate the Product

Now, multiply...\[4.5 \times 4.5 = 20.25\] Therefore, \[(4.5)^2 = 20.25\] Substitute this value into the original expression.
03

Substitute Back

The expression becomes...\[\frac{20.25}{100}\]Use this for further simplification.
04

Simplify the Fraction

Simplify \(\frac{20.25}{100}\) by dividing both the numerator and the denominator by 100. This step involves converting the number into a decimal form:\[20.25 \div 100 = 0.2025\]

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Key Concepts

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

Exponentiation
Exponentiation is essentially a fancy way of saying that you're multiplying a number by itself a certain number of times. In the given problem, we see \( (4.5)^2 \), which means we multiply 4.5 by itself鈥攐nly twice, because 2 is the exponent.
When you perform the multiplication, you should carefully calculate:
  • Multiply the number: In our example, multiply 4.5 by 4.5: \[ 4.5 imes 4.5 \]
  • The result is 20.25: This means that \( (4.5)^2 = 20.25 \).
Exponentiation helps us understand how an initial value grows when multiplied by itself. It's used in everything from calculating areas in geometry to computing complex interest in finance. Always check your results, as each step is crucial!
Fraction Simplification
Simplifying fractions makes calculations easier and more intuitive. Here, we start with \( rac{20.25}{100} \).
Fraction simplification boils down to finding the largest number that divides both the top (numerator) and bottom (denominator) without leaving a remainder. For our fraction, it's clear, as we are dividing by 100.
  • The numerator, 20.25, represents the result from our exponentiation work.
  • The denominator, 100, is a simple power of ten, making it straightforward to simplify.
To simplify \( rac{20.25}{100} \): divide both the numerator and the denominator by 100. Fraction simplification often leads to decimal conversion which we'll explore next, but remember there are no universal rules鈥攐nly logical steps. Aim to make numbers manageable!
Decimal Conversion
Decimal conversion is the process of transforming a fraction into its decimal form. Our last step was about simplifying \( rac{20.25}{100} \). When you perform the division, it takes us smoothly from a fraction to a decimal.
Here's how the conversion works:
  • Perform the division: divide 20.25 by 100.
  • This equals 0.2025, which is the decimal form of our simplified fraction.
Decimal forms are practical because they precisely show the size of the number compared to one whole. In scientific and financial contexts, decimals are often preferred鈥攖hey're less abstract and more user-friendly. When working through problems, always consider if a decimal is more useful for your solution.

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