$6^2
Simplify $\left(2^{-6}\right)(5)\left(2^3 \cdot 5\right)^2$.
Answer (1)
Calculate 6 2 c d o t 3 = 36 c d o t 3 = 108 .
Apply the exponent rule ( ab ) n = a n b n to get ( 2 3 c d o t 5 ) 2 = ( 2 3 ) 2 c d o t 5 2 .
Use the exponent rule ( a m ) n = a mn to get ( 2 3 ) 2 = 2 6 .
Simplify ( 2 − 6 ) ( 5 ) ( 2 6 ) ( 5 2 ) = ( 2 − 6 c d o t 2 6 ) ( 5 c d o t 5 2 ) = ( 1 ) ( 5 c d o t 25 ) = 125 . The final answer is 125 .
Explanation
Understanding the Problem We are asked to simplify two expressions. The first one is 6 2 c d o t 3 , and the second one is ( 2 − 6 ) ( 5 ) ( 2 3 c d o t 5 ) 2 .
Simplifying the First Expression Let's simplify the first expression: 6 2 c d o t 3 = 36 c d o t 3 = 108 .
Applying the Power of a Product Rule Now, let's simplify the second expression: ( 2 − 6 ) ( 5 ) ( 2 3 c d o t 5 ) 2 . First, we apply the exponent rule ( ab ) n = a n b n to the term ( 2 3 c d o t 5 ) 2 , which gives us ( 2 3 ) 2 c d o t 5 2 .
Applying the Power of a Power Rule Next, we use the exponent rule ( a m ) n = a mn to simplify ( 2 3 ) 2 , which gives us 2 3 c d o t 2 = 2 6 . So, the expression becomes ( 2 − 6 ) ( 5 ) ( 2 6 ) ( 5 2 ) .
Combining Powers of 2 Now, we use the exponent rule a m a n = a m + n to combine the powers of 2: 2 − 6 c d o t 2 6 = 2 − 6 + 6 = 2 0 = 1 . So, the expression simplifies to ( 1 ) ( 5 ) ( 5 2 ) = 5 c d o t 25 = 125 .
Final Simplification Therefore, the simplified expression is 125.
Examples
Understanding how to simplify expressions with exponents is crucial in many fields, such as physics and engineering, where you often deal with very large or very small numbers. For example, when calculating the energy of a photon, you use Planck's constant, which is a very small number with a negative exponent. Simplifying expressions allows you to perform calculations more efficiently and accurately, whether you're working with scientific notation or complex equations.
