$\frac{5 \times 2^m-4 \times 2^{m-2}}{3 \times 2^{m+2}-5 \times 2^{m+1}}$
Answer (2)
Rewrite the terms using properties of exponents.
Factor out the common term 2 m from the numerator and denominator.
Simplify the constants in the numerator and denominator.
Cancel out the common factor and simplify the fraction to get the final answer: 2 .
Explanation
Understanding the Problem We are given the expression 3 × 2 m + 2 − 5 × 2 m + 1 5 × 2 m − 4 × 2 m − 2 Our goal is to simplify this expression.
Rewriting with Exponent Properties First, let's rewrite the terms in the numerator and denominator using properties of exponents. Recall that 2 m − 2 = 2 m × 2 − 2 = 2 2 2 m = 4 2 m and 2 m + 2 = 2 m × 2 2 = 4 × 2 m , 2 m + 1 = 2 m × 2 1 = 2 × 2 m . Substituting these into the expression, we get 3 × 4 × 2 m − 5 × 2 × 2 m 5 × 2 m − 4 × 4 2 m
Simplifying Constants Now, we simplify the constants in the numerator and denominator: 12 × 2 m − 10 × 2 m 5 × 2 m − 1 × 2 m
Factoring Out Common Term Next, we factor out 2 m from both the numerator and the denominator: 2 m ( 12 − 10 ) 2 m ( 5 − 1 )
Canceling Common Factor Now, we cancel out the common factor 2 m :
12 − 10 5 − 1 = 2 4
Final Simplification Finally, we simplify the fraction: 2 4 = 2 So, the simplified expression is 2.
Examples
Exponential expressions like the one we simplified are used in various fields, such as calculating compound interest, modeling population growth, and understanding radioactive decay. For instance, if you invest money with a compound interest rate, the formula involves exponential terms similar to those in our expression. Simplifying such expressions helps in making quick estimations and understanding the underlying relationships.
The simplified expression of 3 × 2 m + 2 − 5 × 2 m + 1 5 × 2 m − 4 × 2 m − 2 is 2. This simplification involved rewriting terms using properties of exponents, factoring out common terms, and canceling the common factor. The final result is 2 .
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