Use the properties of exponents to simplify the expression [tex]2\left(x^{\frac{1}{3}}\right)^{\frac{6}{5}} \cdot y^{\frac{2}{3}}[/tex].
Answer (1)
Use the power of a power rule: ( x a ) b = x a ⋅ b .
Multiply the exponents: 3 1 ⋅ 5 6 = 5 2 .
Substitute the simplified exponent back into the expression.
The simplified expression is 2 x 5 2 ⋅ y 3 2 .
Explanation
Understanding the Problem We are asked to simplify the expression 2 ( x 3 1 ) 5 6 c d o t y 3 2 using the properties of exponents.
Applying the Power of a Power Rule To simplify the expression, we need to use the power of a power rule, which states that ( x a ) b = x a c d o t b . In our case, we have ( x 3 1 ) 5 6 , so we need to multiply the exponents 3 1 and 5 6 .
Multiplying the Exponents Let's multiply the exponents: 3 1 c d o t 5 6 = 3 c d o t 5 1 c d o t 6 = 15 6 Now, we simplify the fraction 15 6 by dividing both the numerator and the denominator by their greatest common divisor, which is 3: 15 6 = 15 d i v 3 6 d i v 3 = 5 2
Substituting the Simplified Exponent Now we substitute the simplified exponent back into the expression. We have 2 x 5 2 c d o t y 3 2 . This expression is now simplified because we have applied the power of a power rule and simplified the resulting exponent.
Final Answer Therefore, the simplified expression is 2 x 5 2 c d o t y 3 2 .
Examples
Understanding exponents is crucial in various fields, such as calculating compound interest or modeling exponential growth and decay. For instance, if you invest money in an account with compound interest, the formula involves exponents. Similarly, in biology, exponents are used to model population growth or the decay of radioactive substances. Simplifying expressions with exponents helps in making these calculations more manageable and understandable.
