Use the properties of exponents to rewrite the expression. $(-7 x^4 y^4)(-7 x y^4)$
A. $49 x^4 y^{16}$
B. $49 x^5 y^8$
C. $-49 x^5 y^8$
D. $-49 x^4 y^4$
Answer (2)
Multiply the coefficients: − 7 \t × − 7 = 49 .
Multiply the x terms: x 4 \t × x = x 4 + 1 = x 5 .
Multiply the y terms: y 4 \t × y 4 = y 4 + 4 = y 8 .
Combine the results to get the final answer: 49 x 5 y 8 .
Explanation
Understanding the Problem We are asked to rewrite the expression ( − 7 x 4 y 4 ) ( − 7 x y 4 ) using the properties of exponents. Let's break this down step by step.
Multiplying the Coefficients First, we multiply the coefficients: − 7 × − 7 = 49 . Remember that a negative times a negative is a positive.
Multiplying the x Terms Next, we multiply the x terms. We have x 4 × x . Recall that x is the same as x 1 . Using the property of exponents that x m × x n = x m + n , we get x 4 × x 1 = x 4 + 1 = x 5 .
Multiplying the y Terms Now, we multiply the y terms. We have y 4 × y 4 . Using the same property of exponents, we get y 4 × y 4 = y 4 + 4 = y 8 .
Combining the Results Finally, we combine all the results: 49 x 5 y 8 . Therefore, the rewritten expression is 49 x 5 y 8 .
Examples
Understanding and applying exponent rules is crucial in various fields, such as physics and engineering, where you often deal with very large or very small numbers. For instance, when calculating the energy of a photon, E = h f , where h is Planck's constant and f is the frequency, you might encounter numbers in scientific notation. Simplifying expressions with exponents helps in these calculations. Also, in computer science, when dealing with memory sizes (kilobytes, megabytes, gigabytes), you're working with powers of 2, and exponent rules become essential for understanding storage capacity and data transfer rates.
To rewrite the expression ( − 7 x 4 y 4 ) ( − 7 x y 4 ) , the simplified result is 49 x 5 y 8 from multiplying coefficients and applying the properties of exponents. Therefore, the correct answer is option B.
;
