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Chapter 5: Problem 255

Simplify. \(\left(\frac{1}{5}\right)^{2}+(1.4)(6.5)\)

Short Answer

Expert verified
9.14

Step by step solution

01

Calculate the square of the fraction

First, square the fraction \(\frac{1}{5}\). This means we need to multiply the fraction by itself: \(\frac{1}{5} \times \frac{1}{5} = \frac{1}{25}\).
02

Multiply the decimals

Next, multiply the two decimals 1.4 and 6.5. Perform the multiplication: \(1.4 \times 6.5 = 9.1\).
03

Add the results

Finally, add the results obtained from the previous steps: \(\frac{1}{25} + 9.1\). Converting \(\frac{1}{25}\) to a decimal gives 0.04. Now add the decimal values: \(0.04 + 9.1 = 9.14\).

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Key Concepts

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

Fraction Operations
Understanding fraction operations is essential in many areas of math. To simplify a fraction, you often need to perform multiplication or division. For example, squaring a fraction means multiplying it by itself.

In the problem, we have \(\frac{1}{5}\). Squaring it involves multiplying \(\frac{1}{5}\) by \(\frac{1}{5}\):
\[\frac{1}{5} \times \frac{1}{5} = \frac{1}{25}\].
To multiply fractions, multiply the numerators (top numbers) and the denominators (bottom numbers). The square operation frequently comes up with exponents.

Recap of steps to multiply two fractions:
  • Multiply the numerators together.
  • Multiply the denominators together.
  • Simplify the fraction if possible.
Remember, a simplified fraction is one where the numerator and denominator have no common factors other than 1.
Decimal Multiplication
Decimal multiplication follows regular multiplication rules but involves a step to place the decimal point correctly.

Consider the example from the problem: multiplying 1.4 by 6.5.
To do this:
  • First, ignore the decimals and multiply the numbers as if they were whole numbers: 14 * 65 = 910.
  • Count the total number of decimal places in the original numbers (1 in 1.4 and 1 in 6.5, making a total of 2).
  • Place the decimal point in the answer so that it has the same number of decimal places (2 places), resulting in 9.10 or simply 9.1.
Always double-check that the decimal places in the result match the total from the original numbers.
This ensures your answer is accurate!
Adding Decimals
Adding decimals is straightforward if you align the decimal points correctly.
In the exercise, after converting \(\frac{1}{25}\) to 0.04, you add it to 9.1.
Steps to add decimals:
  • Write the numbers so that the decimal points line up vertically.
  • Add digits starting from the rightmost column, just like you do with whole numbers.
  • If any column sums to 10 or more, carry over to the next column on the left.
For example, 0.04 + 9.1:
  • Write 0.04 and 9.1 with decimals aligned.
  • 0.04
  • + 9.10
The sum is 9.14.
Aligning decimals properly helps ensure that each digit is added to the correct place, yielding an accurate result.

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