Problem 103 - Algebra and Trigonometry

Question

In Exercises 103–106, determine whether each statement makes sense or does not make sense, and explain your reasoning. Knowing the difference between factors and terms is important: In \(\left(3 x^{2} y\right)^{2},\) I can distribute the exponent 2 on each factor, but in \(\left(3 x^{2}+y\right)^{2},\) I cannot do the same thing on each term.

Step-by-Step Solution

Answer

The statement makes sense. It correctly applies the law of exponents: individual terms in a product can each be raised to an exponent, but in a sum, the sum as a whole must be raised to the exponent.

Step 1: Interpret the first expression

In the first expression \((3x^2y)^2\), this is a case of power of a product, and so according to the law of exponents, we can distribute the power 2 to each factor in the expression. Hence, we have \(3^2 * (x^2)^2 * y^2 = 9x^4y^2\)

Step 2: Interpret the second expression

In the second expression \((3x^2+y)^2\), this is a case of power of a sum, and so according to the law of exponents, we cannot distribute the power 2 equally to each term in the expression. Instead, the expression should be squared as a whole, which will result in \((3x^2)^2 + 2*(3x^2)*y + y^2 = 9x^4 + 6x^2y + y^2\)

Step 3: Conclude the interpretation

Therefore, the given statement is correct in terms of the laws of exponents. In the first expression, the power can be distributed to each factor, but in the second expression, the whole expression needs to be squared, not each term separately.