Use the laws of exponents to simplify.
$5^{\frac{3}{5}} \cdot 5^{\frac{1}{10}}$
Answer (1)
Use the property a m ⋅ a n = a m + n .
Add the exponents: 5 3 + 10 1 = 10 6 + 10 1 = 10 7 .
Simplify the expression: 5 5 3 ⋅ 5 10 1 = 5 10 7 .
The simplified expression is 5 10 7 .
Explanation
Understanding the problem We are asked to simplify the expression 5 5 3 ⋅ 5 10 1 using the laws of exponents.
Applying the exponent rule To simplify this expression, we need to use the property of exponents that states a m ⋅ a n = a m + n , where a is the base and m and n are the exponents. In this case, the base is 5, and the exponents are 5 3 and 10 1 .
Adding the exponents So, we have 5 5 3 ⋅ 5 10 1 = 5 5 3 + 10 1 . Now we need to add the fractions in the exponent. To do this, we need to find a common denominator for 5 3 and 10 1 . The least common denominator for 5 and 10 is 10.
Calculating the sum of exponents We can rewrite 5 3 as 10 6 by multiplying both the numerator and the denominator by 2: 5 3 = 5 × 2 3 × 2 = 10 6 . Now we can add the fractions: 10 6 + 10 1 = 10 6 + 1 = 10 7 .
Final simplification Therefore, 5 5 3 ⋅ 5 10 1 = 5 10 7 .
Examples
Understanding exponents is crucial in many fields, such as finance when calculating compound interest. For example, if you invest 1000 a t anann u a l in t eres t r a t eo f 5 A = P(1 + \frac{r}{n})^{nt} , w h ere P$ is the principal amount ( 1000 ) , r i s t h e ann u a l in t eres t r a t e ( 0.05 ) , n i s t h e n u mb ero f t im es t h e in t eres t i sco m p o u n d e d p erye a r ( 4 ) , an d t$ is the number of years. Exponents help determine the growth of your investment over time.
