Simplify the expression: $-(x^2)^3 \cdot x^3$

Apply the power of a power rule: ( x 2 ) 3 = x 2 × 3 = x 6 .
Substitute the result into the original expression: − ( x 2 ) 3 ⋅ x 3 = − x 6 ⋅ x 3 .
Apply the product of powers rule: − x 6 ⋅ x 3 = − x 6 + 3 = − x 9 .
The simplified expression is − x 9 ​ .

Explanation

Understanding the problem We are asked to simplify the expression − ( x 2 ) 3 ". x 3 . This involves using exponent rules to simplify the expression.

Applying the power of a power rule First, we need to simplify the term ( x 2 ) 3 . According to the power of a power rule, ( a m ) n = a m ". n . Applying this rule, we get ( x 2 ) 3 = x 2".3 = x 6 .

Substituting back into the expression Now, substitute x 6 back into the original expression: − ( x 2 ) 3 ". x 3 = − x 6 ". x 3 .

Applying the product of powers rule Next, we simplify − x 6 ". x 3 . According to the product of powers rule, a m ". a n = a m + n . Applying this rule, we get − x 6 ". x 3 = − x 6 + 3 = − x 9 .

Final Answer Therefore, the simplified expression is − x 9 .


Examples
Exponent rules are fundamental in various scientific and engineering calculations. For example, when calculating the volume of a cube with side length x , and then cubing that volume, you would use the power of a power rule. Similarly, in physics, when dealing with quantities that scale with powers of a variable, such as the intensity of light as a function of distance, these rules become essential for simplifying complex expressions and understanding the relationships between different physical quantities. Understanding these rules allows for efficient manipulation and simplification of equations, making complex problems more manageable.

Answered by GinnyAnswer | 2025-07-03

The expression − ( x 2 ) 3 ⋅ x 3 simplifies to − x 9 by applying the power of a power rule and the product of powers rule. First, ( x 2 ) 3 simplifies to x 6 , and then combining − x 6 and x 3 gives − x 9 . Thus, the final answer is − x 9 ​ .
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Answered by Anonymous | 2025-07-04