What value of $n$ makes the statement true?
[tex]6 x^n \cdot 4 x^2=24 x^6[/tex]
[tex]n=\square[/tex]
Answer (1)
Multiply the coefficients: 6 c d o t 4 = 24 .
Apply the exponent rule: x n c d o t x 2 = x n + 2 .
Equate the exponents: n + 2 = 6 .
Solve for n : n = 6 − 2 = 4 . The final answer is 4 .
Explanation
Understanding the Problem We are given the equation 6 x n c d o t 4 x 2 = 24 x 6 and we need to find the value of n that makes the equation true. Let's analyze the equation and use the properties of exponents to solve for n .
Multiplying Coefficients First, we can multiply the coefficients on the left side of the equation: 6 c d o t 4 = 24 . So the equation becomes 24 x n c d o t x 2 = 24 x 6 .
Applying Exponent Rule Next, we use the exponent rule x a c d o t x b = x a + b to simplify the left side of the equation: x n c d o t x 2 = x n + 2 . Now the equation is 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 of x : n + 2 = 6 .
Solving for n Now, we solve for n by subtracting 2 from both sides of the equation: n = 6 − 2 . Therefore, n = 4 .
Final Answer The value of n that makes the statement true is 4. So, the final answer is 4 .
Examples
Imagine you are designing a rectangular garden where the area is determined by the product of its length and width. If the length is represented by x n and the width by x 2 , and you know the total area should be x 6 , finding the correct value of n helps you determine the dimensions of the garden. This problem demonstrates how understanding exponents and algebraic equations can be applied to practical scenarios involving area and dimensions.
