Question

use the properties of exponents to rewrite the expression. $-5y^{2} cdot 5y^{2} cdot 5y^{2} cdot 5y^{2}$
a. $-5y^{8}$
b. $-(5y^{2})^{4}$
c. $(-5y^{2})^{4}$
d. none of these
reset selection

Explanation:

Step1: Recall the property of exponents for multiplication of like bases

When multiplying expressions with the same base, we can use the property \(a^m\cdot a^n=a^{m + n}\), and for a product raised to a power \((ab)^n=a^n b^n\) and \((a^m)^n=a^{mn}\). Also, when we have \(n\) times the multiplication of the same factor \(a\), it can be written as \(a^n\).

The given expression is \(-5y^{2}\cdot5y^{2}\cdot5y^{2}\cdot5y^{2}\). Let's first look at the coefficients and the variable parts separately.

The coefficient part: we have \(- 5\times5\times5\times5\). The variable part: \(y^{2}\times y^{2}\times y^{2}\times y^{2}\)

Step2: Analyze the coefficient and variable parts

For the variable part, using the property \(a^m\cdot a^n=a^{m + n}\), with \(a = y\), \(m=n = 2\) and we are multiplying 4 times. So \(y^{2}\cdot y^{2}\cdot y^{2}\cdot y^{2}=y^{2 + 2+2 + 2}=y^{8}\)

For the coefficient part: \(-5\times5\times5\times5=- (5\times5\times5\times5)\). Since \(5\times5\times5\times5 = 5^{4}\), the coefficient part is \(-5^{4}\)

Now, the entire expression can be rewritten by combining the coefficient and variable parts. Let's look at the factors: the first factor is \(-5y^{2}\) and the other three factors are \(5y^{2}\). Wait, actually, let's count the number of times \(5y^{2}\) is multiplied with a negative sign at the beginning.

The expression is \(-5y^{2}\times(5y^{2})\times(5y^{2})\times(5y^{2})\). Let's factor out the negative sign: \(-\times(5y^{2})\times(5y^{2})\times(5y^{2})\times(5y^{2})\)

Since we have 4 factors of \(5y^{2}\) multiplied together and then multiplied by - 1, we can write this as \(-(5y^{2})^{4}\)

Let's check option A: \(-5y^{8}\). The coefficient here is - 5, but in our original expression, the coefficient of \(y^{8}\) should be \(-5\times5\times5\times5=-5^{4}=-625\), while \(-5y^{8}\) has a coefficient of - 5, so A is wrong.

Option C: \((-5y^{2})^{4}=(-5)^{4}(y^{2})^{4}=5^{4}y^{8}\), which is positive, but our original expression has a negative sign (because the first coefficient is - 5 and the rest are positive 5s, so the overall coefficient is negative), so C is wrong.

Option B: \(-(5y^{2})^{4}=-(5^{4}(y^{2})^{4})=-5^{4}y^{8}\), which matches the expression we derived from the original: \(-5y^{2}\cdot5y^{2}\cdot5y^{2}\cdot5y^{2}=- (5y^{2})^{4}\)

Answer:

B. \(-(5y^{2})^{4}\)