Write $10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10$ with an exponent.
The problem asks to represent the product of seven tens, \(10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10,\) as a single power of 10. Exponentiation is a shorthand notation for repeated multiplication. When a number \(a\) is multiplied by itself \(n\) times, we express this as \(a^n\), where \(a\) is called the base and \(n\) is called the exponent. In our case, the number being multiplied is 10, so our base is 10.
Next, we count how many times 10 appears as a factor in our expression:
\[10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = \underbrace{10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10}_{7 \text{ times}}\]
Since the base 10 is multiplied by itself 7 times, the exponent is 7. Therefore, the expression can be written as \(10^7\).
Exponents are incredibly useful in science and everyday life to express very large numbers concisely. For instance, the distance from the Earth to the Sun is approximately 150,000,000 kilometers. Instead of writing all those zeros, scientists write this as \(1.5 \times 10^8\) km. Using exponents helps us handle massive scales, like measuring the speed of light or the number of stars in the galaxy, without getting lost in strings of zeros.