Question

How many digits does the \(10^{120}\) th power of \(10^{100}\) have?

Short answer

The number has \(10^{122} + 1\) digits.

Step-by-step solution

Understanding the Expression

The expression given in the problem is \((10^{100})^{10^{120}}\). We need to simplify this expression first.

Applying Power Rule

Use the power rule for exponents which states \((a^m)^n = a^{m imes n}\). Here, \(a = 10\), \(m = 100\), and \(n = 10^{120}\). Thus, \((10^{100})^{10^{120}} = 10^{100 imes 10^{120}}\).

Calculating the Exponent

Simplify the exponent: \(100 \times 10^{120} = 10^{2} \times 10^{120} = 10^{122}\). Thus the expression reduces to \(10^{10^{122}}\).

Understanding Powers of 10

The expression \(10^{10^{122}}\) indicates a number with 1 followed by \(10^{122}\) zeros, which means the number has \(10^{122} + 1\) digits.

Key concepts

Power Rule

Exponentiation is a powerful mathematical concept, and the power rule is one of its fundamental properties. It allows us to simplify expressions with exponents. In general, the power rule states that for any base \(a\) and exponents \(m\) and \(n\), \((a^m)^n = a^{m \times n}\). This means that when you raise a power to another power, you simply multiply the exponents together.

Let's see how this applies to our problem. We're given the expression \((10^{100})^{10^{120}}\). Here, the base \(a\) is 10, the first exponent \(m\) is 100, and the exponent of the exponent \(n\) is \(10^{120}\).

• Using the power rule, we multiply these exponents: \[100 \times 10^{120}\].
• This simplifies further to \(10^2 \times 10^{120}\), which is \(10^{122}\).

This simplification is essential for reducing complex expressions, especially when working with large numbers in powers of ten.

Digits Calculation

Once you have a simplified power expression, like \(10^{10^{122}}\), calculating the number of digits becomes straightforward. Numbers in the form of \(10^n\) have a very specific characteristic. They all have \(n+1\) digits.

Why is this the case? Consider the number \(10^2\), which is 100. It has three digits. Similarly, \(10^3\) is 1000, and has four digits. In each instance, the number of digits is one more than the exponent.

• For our expression \(10^{10^{122}}\), this showcases a number starting with 1 followed by \(10^{122}\) zeros.
• This means the number contains \(10^{122} + 1\) digits in total.

Understanding this rule helps quickly determine the digits of extremely large numbers in exponential form without calculating them.

Simplifying Expressions

Being able to simplify expressions, particularly those involving exponents, is a crucial mathematical skill. It reduces complexity, making problems easier to solve and understand.

For instance, consider our initial expression \((10^{100})^{10^{120}}\). With large numbers, such simplification is not just helpful—it’s often necessary. Here’s a clear approach:

• Apply the power rule: this operation will condense the multiple exponents into a single, simpler figure.
• Recognize that powers of 10, like \(10^{10^{122}}\), have intuitive properties related to digit calculation.

These strategies help tackle even the most intimidating numbers by focusing on exponent rules and the unique properties of powers of 10. This understanding bridges theoretical exercises and real-world applications effectively.