The population of a city is 45,000 and decreases 2% each year. If the trend continues, what will the population be after 15 year
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1) Change radical forms to fractional exponents using the rule:
The n<span>th root of "</span>a number" = "that number" raised to the<span> reciprocal of n.
For example </span>![\sqrt[n]{3} = 3^{ \frac{1}{n} }](https://tex.z-dn.net/?f=%20%5Csqrt%5Bn%5D%7B3%7D%20%3D%20%20%203%5E%7B%20%5Cfrac%7B1%7D%7Bn%7D%20%7D)
.
The square root of 3 (
) = 3 to the one-half power (
).
The 5th root of 3 (![\sqrt[5]{3}](https://tex.z-dn.net/?f=%20%5Csqrt%5B5%5D%7B3%7D%20)
) = 3 to the one-fifth power (
).
2) Now use the product of powers exponent rule to simplify:
This rule says
. When two expressions with the same base (a, in this example) are multiplied, you
can add their exponents while keeping the same base.
You now have
. These two expressions have the same base, 3. That means you can add their exponents:
3) You can leave it in the form
or change it back into a radical ![\sqrt[10]{3^7}](https://tex.z-dn.net/?f=%20%5Csqrt%5B10%5D%7B3%5E7%7D%20)
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Answer: or
The n<span>th root of "</span>a number" = "that number" raised to the<span> reciprocal of n.
For example </span>
The square root of 3 (
The 5th root of 3 (
2) Now use the product of powers exponent rule to simplify:
This rule says
You now have
3) You can leave it in the form
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Answer: