Find the value of the expression:
$\frac{3^7 \cdot 27}{\left(3^4\right)^3}$
Answer (2)
Rewrite 27 as 3 3 .
Simplify the numerator: 3 7 × 3 3 = 3 10 .
Simplify the denominator: ( 3 4 ) 3 = 3 12 .
Simplify the fraction: 3 12 3 10 = 3 − 2 = 9 1 .
Explanation
Understanding the Problem We are asked to find the value of the expression ( 3 4 ) 3 3 7 × 27 . This involves simplifying an expression with exponents. We will use the properties of exponents to simplify the expression.
Rewriting 27 as a Power of 3 First, we rewrite 27 as a power of 3: 27 = 3 3 . So the expression becomes ( 3 4 ) 3 3 7 × 3 3 .
Simplifying the Numerator Next, we simplify the numerator using the rule a m × a n = a m + n : 3 7 × 3 3 = 3 7 + 3 = 3 10 .
Simplifying the Denominator Now, we simplify the denominator using the rule ( a m ) n = a m × n : ( 3 4 ) 3 = 3 4 × 3 = 3 12 .
Simplifying the Fraction The expression is now 3 12 3 10 . We simplify this using the rule a n a m = a m − n : 3 12 3 10 = 3 10 − 12 = 3 − 2 .
Calculating the Final Value Finally, we rewrite 3 − 2 using the rule a − n = a n 1 : 3 − 2 = 3 2 1 = 9 1 .
Examples
Understanding exponents is crucial in many fields, such as 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 t ye a rs i s g i v e nb y A = 1000(1 + 0.05)^t . S im pl i f y in g t hi se x p ress i o n f or d i ff ere n t v a l u eso f t$ involves using the properties of exponents, similar to the problem we just solved. This concept is also used in calculating population growth, radioactive decay, and many other real-world phenomena.
To simplify the expression ( 3 4 ) 3 3 7 ⋅ 27 , we rewrite 27 as 3 3 and simplify to get 3 12 3 10 = 3 − 2 = 9 1 . Therefore, the final value is 9 1 .
;
