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Write a power represented with a positive base and a positive exponent whose value is less than the base.

Write a power represented with a positive base and a positive exponent whose value is less than the base.

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Properties of logarithms: mixed review

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Q. Write a power represented with a positive base and a positive exponent whose value is less than the base.

Understand the problem: Understand the problem. $\newline$ We need to find a power with a positive base and a positive exponent such that the value of the power is less than the base. This means we are looking for an expression of the form $b^e$ where $b > 1$ , $e > 0$ , and $b^e < b$ .
Choose a base: Choose a base greater than $1$ . Let's choose a simple base, for example, $b = 2$ , which is a positive integer greater than $1$ .
Choose an exponent: Choose an exponent between $0$ and $1$ . Since we want the value of the power to be less than the base, we need an exponent that is between $0$ and $1$ (not including $0$ ). Let's choose $e = 0.5$ , which is a positive number less than $1$ .
Calculate the power: Calculate the power. $\newline$ Now we calculate $2^{0.5}$ , which is the square root of $2$ . The square root of $2$ is approximately $1.414$ , which is less than the base $2$ .
Verify the value: Verify that the value of the power is less than the base. $\newline$ Since $1.414$ (the value of $2^{0.5}$ ) is less than $2$ (the base), we have found a power that meets the criteria.

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