Question

Express each of the following powers of 10 as an ordinary number: (a) \(1 \times 10^{\text {I }}\) (b) \(1 \times 10^{-1}\)

Short answer

(a) 10, (b) 0.1

Step-by-step solution

Understand Powers of 10

The expression \(10^n\) means \(10\) raised to the power of \(n\). If \(n\) is positive, \(10^n\) is \(1\) followed by \(n\) zeros. If \(n\) is negative, \(10^n\) is \(1\) divided by \(10^n\).

Evaluate \(1 \times 10^{1}\)

Since \(10^1\) means 10 raised to the power of 1, which is simply \(10\). Thus the expression \(1 \times 10^1 = 10\).

Evaluate \(1 \times 10^{-1}\)

For \(10^{-1}\), it represents \(1\) divided by \(10\), which is \(0.1\), since any negative power of 10 shifts the decimal point to the left. Thus, \(1 \times 10^{-1}\) equals \(0.1\).

Key concepts

Understanding Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact form. It involves two parts: a coefficient, usually a number between 1 and 10, and a power of 10. This makes it easier to manage and read lengthy numbers.
The format is:

• A number between 1 and 10
• Multiplied by a power of 10 (e.g., \(10^n\))

This notation is helpful when dealing with numbers like the following:

• Very large numbers, e.g., \(5,000,000 = 5 \times 10^6\)
• Very small numbers, e.g., \(0.000005 = 5 \times 10^{-6}\)

The power of 10 indicates how many places we move the decimal point. A positive exponent means to move it to the right, and a negative exponent means to move it to the left.

Negative Exponents Explained

Negative exponents represent a division rather than a multiplication. For the power of 10, each negative exponent indicates how many times the number 10 should be used in division.
Here are a few key points about negative exponents:

• \(10^{-1} = \frac{1}{10}\), which equals 0.1
• \(10^{-2} = \frac{1}{10^2} = \frac{1}{100}\), which equals 0.01
• The higher the absolute value of the exponent, the smaller the number becomes

Take the number \(10^{-3}\). This means 1 divided by \(10^3\), resulting in \(0.001\).
Understanding this concept helps make sense of small decimal quantities, which appear frequently in scientific calculations and financial data.

Positive Exponents Clarified

When dealing with positive exponents, the concept is straightforward. Raising a number to a positive exponent involves repeated multiplication.
Here are the essentials of positive exponents:

• \(10^1 = 10\), meaning "10 to the power of 1" is just 10.
• \(10^2 = 10 \times 10 = 100\), indicating we multiply 10 by itself once.
• Larger exponents mean larger numbers, like \(10^3 = 1000\).

With positive exponents, the number grows quickly. This principle is useful in representing large values succinctly, such as population sizes or distances in astronomy.
Remember, each increase in the exponent shifts the decimal point one place to the right, simplifying the multiplication of large numbers.