Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Nov 13, 2023

Use properties of integer exponents to explain the meaning of $2^{\frac{1}{5}}$ .

STEP 1

Assumptions1. We are given that $^{\frac{1}{5}} \cdot^{\frac{1}{5}} \cdot^{\frac{1}{5}} \cdot^{\frac{1}{5}} \cdot^{\frac{1}{5}}=$ . . We are asked to explain the meaning of $^{\frac{1}{5}}$ using the properties of integer exponents.

STEP 2

The property of integer exponents states that when you multiply numbers with the same base, you add the exponents. In this case, the base is2 and the exponent is $\frac{1}{5}$ .
$2^{\frac{1}{5}} \cdot2^{\frac{1}{5}} \cdot2^{\frac{1}{5}} \cdot2^{\frac{1}{5}} \cdot2^{\frac{1}{5}} =2^{\frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5}}$

STEP 3

Now, add the exponents.
$2^{\frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5}} =2^{5 \cdot \frac{1}{5}}$

STEP 4

implify the exponent.
$2^{ \cdot \frac{1}{}} =2^1$

STEP 5

We know that any number raised to the power of1 is the number itself, so $2^1 =2$ .

STEP 6

We are given that $2^{\frac{1}{5}} \cdot2^{\frac{1}{5}} \cdot2^{\frac{1}{5}} \cdot2^{\frac{1}{5}} \cdot2^{\frac{1}{5}}=2$ , and we have shown that this is equal to $2^1$ . Therefore, $2^{\frac{1}{5}}$ is the fifth root of2, because when you multiply it by itself five times, you get2.
In conclusion, $2^{\frac{1}{5}}$ represents the fifth root of2.

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