Simplify the expression: $4\left(x^2\right)^4 \cdot(-x)^3$
Answer (1)
Apply the power of a power rule: ( x 2 ) 4 = x 8 .
Simplify the negative term: ( − x ) 3 = − x 3 .
Substitute the simplified terms back into the original expression: 4 ( x 8 ) ⋅ ( − x 3 ) .
Multiply the terms together: − 4 x 11 .
Explanation
Understanding the Problem We are given the expression 4 ( x 2 ) 4 ⋅ ( − x ) 3 . Our goal is to simplify this expression by applying exponent rules and combining like terms.
Applying the Power of a Power Rule First, we simplify ( x 2 ) 4 using the power of a power rule, which states that ( x a ) b = x a ⋅ b . In this case, we have ( x 2 ) 4 = x 2 ⋅ 4 = x 8 .
Simplifying the Negative Term Next, we simplify ( − x ) 3 . This means ( − 1 ⋅ x ) 3 = ( − 1 ) 3 ⋅ x 3 . Since ( − 1 ) 3 = − 1 , we have ( − x ) 3 = − x 3 .
Substituting Back into the Expression Now we substitute these simplified terms back into the original expression: 4 ( x 2 ) 4 ⋅ ( − x ) 3 = 4 ( x 8 ) ⋅ ( − x 3 ) .
Multiplying and Combining Like Terms Finally, we multiply the terms together. We have 4 ⋅ x 8 ⋅ ( − 1 ) ⋅ x 3 = 4 ⋅ ( − 1 ) ⋅ x 8 ⋅ x 3 = − 4 ⋅ x 8 + 3 = − 4 x 11 .
Final Answer Therefore, the simplified expression is − 4 x 11 .
Examples
Understanding how to simplify expressions with exponents is crucial in many areas of mathematics and physics. For example, in physics, when dealing with the intensity of light or sound, which decreases with the square of the distance from the source, simplifying expressions involving powers becomes essential. Imagine calculating the power output of a solar panel, where the voltage and current are related by exponents. Simplifying such expressions helps engineers optimize the panel's design for maximum efficiency. This skill also helps in understanding exponential growth and decay, which are fundamental in modeling population dynamics or radioactive decay.
