Question

Express each of the following powers of 10 as an ordinary number: (a) \(1 \times 10^{0}\) (b) \(1 \times 10^{-10}\)

Short answer

(a) 1; (b) 0.0000000001.

Step-by-step solution

Understand the Power of 10

The expression is given as a power of 10. The power indicates how many times to multiply the number by 10. If the exponent is positive, you multiply by 10 that many times. If the exponent is zero, the result is always 1. If the exponent is negative, you divide by 10 that many times.

Solve Part (a)

The expression is given as \(1 \times 10^{0}\). Any number raised to the power of zero is 1. Therefore, \(1 \times 10^{0} = 1 \).

Solve Part (b)

The expression is \(1 \times 10^{-10}\). A negative exponent means we will be dividing by 10 eleven times. This is equivalent to moving the decimal point to the left: \(1 \times 10^{-10} = 0.0000000001\).

Key concepts

Powers of Ten

Understanding powers of ten is fundamental to working with large and small numbers. When you see a power of ten like \(10^3\), it means you multiply the number 10 by itself three times: \(10 \times 10 \times 10 = 1000\). This makes handling large numbers more manageable by expressing them in compact forms. For instance, 1000 can be expressed simply as \(10^3\). Applying powers of ten helps streamline calculations and clarifies the scale we are dealing with, be it thousands, millions, or more.
Conversely, negative powers of ten help us deal with small numbers. They represent division rather than multiplication. Each negative exponent shows how many times to divide by 10. For example, \(10^{-3}\) means \(\frac{1}{10 \times 10 \times 10} = 0.001\). This is useful in scientific notation to efficiently express small values like an electron's size or the frequency of a lightwave. The ability to switch between these forms is crucial in science and mathematics.

Negative Exponents

Negative exponents might seem tricky at first, but they're easier to understand once you see them as simply the inverse of multiplying by ten. When you have a negative exponent, like \(10^{-2}\), you are essentially taking the reciprocal of \(10^2\). In this case, \(10^{-2} = \frac{1}{10^2} = \frac{1}{100} = 0.01\).
To visualize, imagine shifting the decimal point to the left. Each step is like dividing by 10. So, \(10^{-1}\) equals 0.1, \(10^{-2}\) equals 0.01, \(10^{-3}\) equals 0.001, and so on. This is particularly useful when you're dealing with numbers less than one in scientific notation.

• Helpful Tip: The more negative the exponent, the smaller the number.
• For conversion: Shift the decimal left according to the absolute value of the exponent.

Exponent Rules

Exponent rules are like guiding principles for simplifying and solving mathematical expressions with powers. A key rule is when you multiply like bases, you add their exponents: \(a^m \times a^n = a^{m+n}\). For example, \(10^3 \times 10^2 = 10^{3+2} = 10^5\).
Another important rule involves division, where you subtract exponents: \(\frac{a^m}{a^n} = a^{m-n}\). For instance, \(\frac{10^5}{10^2} = 10^{5-2} = 10^3\).
These rules extend to negative exponents as well. If you encounter an expression like \(10^{-2} \times 10^4\), you can apply the multiplication rule: \(10^{-2 + 4} = 10^2\). Likewise, powers raised to other powers, such as \((a^m)^n = a^{m \times n}\), let you multiply exponents. Knowing and using these rules helps you solve complex equations with ease.

• Add exponents when multiplying similar bases.
• Subtract exponents when dividing.
• Remember: a negative exponent means division.