Question

follow the instructions below. write ((y^{2})^{3}) without exponents. (left(y^{2}
ight)^{3}=square) fill in the blank. (left(y^{2}
ight)^{3}=y^{square})

Explanation:

First Part: Write \((y^{2})^{3}\) without exponents

Step1: Recall the power of a power rule

The power of a power rule states that \((a^{m})^{n}=a^{m\times n}\). Here, \(a = y\), \(m = 2\), and \(n = 3\). But we need to write it without exponents, so we expand the exponents. First, \((y^{2})^{3}\) means \(y^{2}\) multiplied by itself 3 times.

Step2: Expand the expression

\((y^{2})^{3}=y^{2}\times y^{2}\times y^{2}\). When multiplying variables with exponents, we add the exponents (since \(y^{a}\times y^{b}=y^{a + b}\)). So, \(2+2+2 = 6\), and \(y^{2}\times y^{2}\times y^{2}=y\times y\times y\times y\times y\times y\) (because \(y^{2}=y\times y\), so multiplying three of those gives six \(y\)s multiplied together).

Step1: Apply the power of a power rule

The power of a power rule is \((a^{m})^{n}=a^{m\times n}\). For \((y^{2})^{3}\), \(a = y\), \(m = 2\), and \(n = 3\).

Step2: Calculate the exponent

Multiply the exponents: \(2\times3 = 6\). So \((y^{2})^{3}=y^{6}\).

Answer:

\(y\times y\times y\times y\times y\times y\)

Second Part: Fill in the blank for \((y^{2})^{3}=y^{\square}\)