What value of $n$ makes the statement true?
$6 x^n \cdot 4 x^2=24 x^6$
$n = \square$
Answer (1)
Multiply the coefficients: 6"."4 = 24 .
Apply the exponent rule: x n "." x 2 = x n + 2 .
Equate the exponents: n + 2 = 6 .
Solve for n : n = 4 . The final answer is 4 .
Explanation
Understanding the Problem We are given the equation 6 x n "."4 x 2 = 24 x 6 . Our goal is to find the value of n that makes this statement true. This involves understanding how to multiply terms with exponents.
Multiplying Coefficients First, let's multiply the coefficients: 6"."4 = 24 . So, the equation becomes 24 x n "." x 2 = 24 x 6 .
Applying the Exponent Rule Next, we use the rule for multiplying exponential terms with the same base: x a "." x b = x a + b . Applying this rule, we have x n "." x 2 = x n + 2 . Thus, the equation is now 24 x n + 2 = 24 x 6 .
Equating Exponents Since the coefficients on both sides of the equation are equal (both are 24), we can equate the exponents: n + 2 = 6 .
Solving for n Now, we solve for n . Subtract 2 from both sides of the equation: n = 6 − 2 . Therefore, n = 4 .
Final Answer So, the value of n that makes the statement true is 4.
Examples
Imagine you are designing a rectangular garden where the area is determined by the expression 6 x n ⋅ 4 x 2 . If you want the area to be 24 x 6 , finding the correct value of n helps you determine the dimensions of the garden. This concept is also used in physics to calculate quantities like kinetic energy or potential energy, where variables are raised to certain powers, and you need to determine the correct exponent to match observed data. Understanding exponents and their properties is crucial for modeling and solving real-world problems in various scientific and engineering fields.
