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Chapter 1: Problem 4

To raise a power of 10 to another power, you multiply the exponent by the power. Explain why this works.

Short Answer

Expert verified
The rule that to raise a power of 10 to another power you multiply the exponents, works because exponentiation is essentially repeated multiplication. When you have \(10^{m}\) raised to 'n', you're multiplying 10 by itself 'm' times and then repeating that 'n' times, which is equivalent to multiplying 10 by itself 'm*n' times.

Step by step solution

01

Understanding the Rule of Exponents

First, it is important to understand the rule of exponents \((a^{m})^{n} = a^{m*n}\). According to this rule, when an exponent is raised to another exponent, the two exponents are multiplied. The base, labeled as 'a' here, remains the same.
02

Application to Powers of 10

When applying this rule to powers of 10, the base, 'a', is 10. So if we have \(10^{m}\) raised to the power of 'n', then according to the rules of exponents, we would multiply 'm' and 'n' and the base would remain 10. Thus, \((10^{m})^{n} = 10^{m*n}\).
03

Reasoning

The reasoning behind this rule is linked to how exponentiation works. Exponentiation is repeated multiplication. When we say \(10^{m}\), we are multiplying 10 by itself 'm' times. When we then raise \(10^{m}\) to the power 'n', we're repeating \(10^{m}\) 'n' times. This is the equivalent of multiplying 10 by itself 'm*n' times, hence the rule \((10^{m})^{n} = 10^{m*n}\).

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