\frac{(-2)^5 \times 2^3 \times(-2)^4}{2 \times(-2)^2 \times 2^2}
Answer (1)
Rewrite all terms as powers of 2, considering the sign.
Substitute these values into the expression and combine the powers of 2 in the numerator and denominator.
Simplify the fraction by subtracting the exponents.
Calculate the final result: − 128
Explanation
Understanding the Problem We are asked to simplify the expression 2 × ( − 2 ) 2 × 2 2 ( − 2 ) 5 × 2 3 × ( − 2 ) 4 . This involves powers of 2 and -2. Our goal is to simplify this expression to a numerical value.
Rewriting the Terms First, let's rewrite all terms as powers of 2, considering the sign. We have ( − 2 ) 5 = − 2 5 , ( − 2 ) 4 = 2 4 , and ( − 2 ) 2 = 2 2 .
Substituting the Values Now, substitute these into the expression: 2 × 2 2 × 2 2 − 2 5 × 2 3 × 2 4 .
Simplifying the Numerator Next, combine the powers of 2 in the numerator: 2 5 + 3 + 4 = 2 12 . So, the numerator becomes − 2 12 .
Simplifying the Denominator Combine the powers of 2 in the denominator: 2 1 + 2 + 2 = 2 5 . Thus, the denominator becomes 2 5 .
Combining the Results The expression is now 2 5 − 2 12 .
Simplifying the Fraction Simplify the fraction by subtracting the exponents: 2 12 − 5 = 2 7 .
Calculating the Power The result is − 2 7 .
Final Answer Finally, calculate 2 7 = 128 . Therefore, the final answer is -128.
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 involves exponents. Simplifying such expressions helps in making accurate calculations and predictions in these fields.
