Question

Express each of the following as an "ordinary" decimal number. a. \(7.327 \times 10^{-4}\) b. \(1.51 \times 10^{2}\) c. \(1 \times 10^{0}\) d. \(5.399 \times 10^{-4}\) e. \(0.221 \times 10^{3}\) f. \(7.83 \times 10^{-2}\) g. \(1218 \times 10^{-4}\) h. \(2.918 \times 10^{-4}\) i. \(7.251 \times 10^{3}\) j. \(1.911 \times 10^{-9}\) k. \(9.951 \times 10^{2}\) 1\. \(9.951 \times 10^{-2}\)

Short answer

a. \(0.0007327\) b. \(151\) c. \(1\) d. \(0.0005399\) e. \(221\) f. \(0.0783\) g. \(0.1218\) h. \(0.0002918\) i. \(7251\) j. \(0.000000001911\) k. \(995.1\) 1. \(0.09951\)

Step-by-step solution

Solution for (a)

Given: \(7.327 \times 10^{-4}\) Since the exponent is -4, move the decimal point 4 places to the left. Answer: \(0.0007327\)

Solution for (b)

Given: \(1.51 \times 10^{2}\) Since the exponent is 2, move the decimal point 2 places to the right. Answer: \(151\)

Solution for (c)

Given: \(1 \times 10^{0}\) Since the exponent is 0, the number remains unchanged. Answer: \(1\)

Solution for (d)

Given: \(5.399 \times 10^{-4}\) Since the exponent is -4, move the decimal point 4 places to the left. Answer: \(0.0005399\)

Solution for (e)

Given: \(0.221 \times 10^{3}\) Since the exponent is 3, move the decimal point 3 places to the right. Answer: \(221\)

Solution for (f)

Given: \(7.83 \times 10^{-2}\) Since the exponent is -2, move the decimal point 2 places to the left. Answer: \(0.0783\)

Solution for (g)

Given: \(1218 \times 10^{-4}\) Since the exponent is -4, move the decimal point 4 places to the left. Answer: \(0.1218\)

Solution for (h)

Given: \(2.918 \times 10^{-4}\) Since the exponent is -4, move the decimal point 4 places to the left. Answer: \(0.0002918\)

Solution for (i)

Given: \(7.251 \times 10^{3}\) Since the exponent is 3, move the decimal point 3 places to the right. Answer: \(7251\)

Solution for (j)

Given: \(1.911 \times 10^{-9}\) Since the exponent is -9, move the decimal point 9 places to the left. Answer: \(0.000000001911\)

Solution for (k)

Given: \(9.951 \times 10^{2}\) Since the exponent is 2, move the decimal point 2 places to the right. Answer: \(995.1\)

Solution for (1)

Given: \(9.951 \times 10^{-2}\) Since the exponent is -2, move the decimal point 2 places to the left. Answer: \(0.09951\)

Key concepts

Decimal Notation

Understanding decimal notation is fundamental in mathematics, especially when working with both large and small numbers. Think of it as a way of writing numbers that include fractions using the base-ten system. The number 123.45, for example, has '123' as the whole number part and '45' as the decimal or fractional part.

In the scientific notation conversion exercises, like expressing a number such as \(7.327 \times 10^{-4}\), we're converting from a scientific format that is concise and easy to write for very large or small numbers, to a more 'ordinary' or extended decimal form, which can be more intuitive to comprehend. The conversion process helps to directly see the value of the number as a fraction of ten, without the need to comprehend the exponent at first glance.

Exponents in Mathematics

An exponent in mathematics tells us how many times a number should be multiplied by itself. It's a compact way to write multiplication that would otherwise take up a lot of space. For example, \(10^2\) is much simpler than writing 10 x 10. In the context of scientific notation, the exponent can be positive or negative. A positive exponent indicates how many times to multiply a number by 10, while a negative exponent indicates how many times the number should be divided by 10.

When converting exponentials like \(1.51 \times 10^{2}\) into decimal notation, you shift the decimal point to the right for positive exponents, in this case giving us 151. Contrastingly, with a negative exponent, such as \(7.327 \times 10^{-4}\), the decimal moves to the left, manifesting as 0.0007327. This is a crucial transformation for appreciating the scale and magnitude of numbers in different contexts.

Place Value

Place value represents the value of each digit in a number based on its position. In our base-ten system, each place represents, in ascending order from right to left, ten times the value of the place to its right. For example, in the number 345, the '5' is in the ones place, '4' in the tens place, and '3' in the hundreds place.

Understanding place value is essential when working with scientific notation conversion. Moving the decimal point changes the place value of the digits. As illustrated in the exercise solution for \(2.918 \times 10^{-4}\), when we move the decimal four places to the left to get 0.0002918, each digit's place value changes from ten-thousandths to units. This understanding helps us accurately interpret and rewrite the numbers in their decimal form, affirming the importance of each digit's role in representing a number.