Question

Express each of the following products as a power of 10 : (a) \(10 \times 10 \times 10 \times 10\) (b) \(1 / 10 \times 1 / 10 \times 1 / 10 \times 1 / 10\)

Short answer

(a) \(10^4\), (b) \(10^{-4}\).

Step-by-step solution

Identify Repeated Multiplication

In part (a), identify the number of times 10 is multiplied by itself: four times. In part (b), identify the number of times \( \frac{1}{10} \) is multiplied by itself: four times as well.

Express as Power of 10

In part (a), the expression \(10 \times 10 \times 10 \times 10\) can be rewritten as \(10^4\) because it is 10 multiplied by itself four times. In part (b), the expression \(\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10}\) can be rewritten as \((\frac{1}{10})^4\) or alternately as \(10^{-4}\), since each \(\frac{1}{10}\) corresponds to \(10^{-1}\) and multiplying \(10^{-1}\) four times gives \(10^{-4}\).

Key concepts

Multiplication

Multiplication is a fundamental mathematical operation used to combine equal groups of objects or numbers. It's often thought of as repeated addition. For example, when you see an expression like \(10 \times 10\), you are essentially adding 10 to itself. Let's break it down a little further:

• \(10 \times 10\): Adds 10 ten times, which equals 100.
• \(10 \times 10 \times 10\): Adds 10 one hundred times, which equals 1000.
• \(10 \times 10 \times 10 \times 10\): Adds 10 one thousand times, which equals 10000.

Understanding the power of multiplication helps simplify expressions, especially when dealing with powers of ten. So, instead of writing \(10 \times 10 \times 10 \times 10\), you can express it as a single term using exponents, like \(10^4\), meaning 10 is used as a factor four times.

Exponents

Exponents are a shorthand way to show how many times a number (the base) is multiplied by itself. It makes things neat and tidy, especially with bigger numbers.

• In an expression like \(a^n\), \(a\) is the base, and \(n\) is the exponent (or power).
• The exponent tells us how many times to multiply the base by itself. For example, \(10^4\) means multiply 10 by itself four times.
• The outcome of \(10^4\) would be \(10 \times 10 \times 10 \times 10 = 10000\).

Exponents simplify expressions by removing the clutter of repeated multiplication. They also help in comparing large numbers more efficiently. Using exponents is especially handy when working with powers of ten due to their base structure, which corresponds perfectly with our decimal system.

Negative Exponents

Negative exponents might look tricky at first, but they simplify a whole new world of dividing powers. A negative exponent indicates division rather than multiplication. Here's how:

• When you see an expression like \(a^{-n}\), it means the reciprocal of the base raised to the positive exponent, \(\frac{1}{a^n}\).
• So, \(10^{-1}\) is equivalent to \(\frac{1}{10}\).
• In cases like the exercise, \(10^{-4}\), this means \(\frac{1}{10 \times 10 \times 10 \times 10}\) or simply \(\frac{1}{10000}\).

Negative exponents allow us to express division in terms of multiplication and unify the concept of exponents with a straightforward rule: any base to a negative power is one over that base to the positive power. This relationship is incredibly useful in scientific notation and when dealing with fractions of powers of ten.