Answer :
To simplify the expression
[tex]$$\left(5xy^5\right)^2\left(y^3\right)^4,$$[/tex]
follow these steps:
1. First, simplify the term [tex]$$\left(5xy^5\right)^2.$$[/tex]
Raising each factor inside the parentheses to the power of 2 gives:
[tex]$$\left(5xy^5\right)^2 = 5^2 \cdot x^2 \cdot \left(y^5\right)^2.$$[/tex]
Since [tex]$5^2 = 25$[/tex] and [tex]$\left(y^5\right)^2 = y^{5 \cdot 2} = y^{10}$[/tex], we have:
[tex]$$\left(5xy^5\right)^2 = 25x^2y^{10}.$$[/tex]
2. Next, simplify the term [tex]$$\left(y^3\right)^4.$$[/tex]
Raising [tex]$y^3$[/tex] to the power of 4 gives:
[tex]$$\left(y^3\right)^4 = y^{3 \cdot 4} = y^{12}.$$[/tex]
3. Now, multiply the results of the two simplifications:
[tex]$$25x^2y^{10} \cdot y^{12}.$$[/tex]
When multiplying powers of the same base, you add the exponents:
[tex]$$y^{10} \cdot y^{12} = y^{10+12} = y^{22}.$$[/tex]
Thus, the product becomes:
[tex]$$25x^2y^{22}.$$[/tex]
The simplified expression is
[tex]$$\boxed{25x^2y^{22}}.$$[/tex]
[tex]$$\left(5xy^5\right)^2\left(y^3\right)^4,$$[/tex]
follow these steps:
1. First, simplify the term [tex]$$\left(5xy^5\right)^2.$$[/tex]
Raising each factor inside the parentheses to the power of 2 gives:
[tex]$$\left(5xy^5\right)^2 = 5^2 \cdot x^2 \cdot \left(y^5\right)^2.$$[/tex]
Since [tex]$5^2 = 25$[/tex] and [tex]$\left(y^5\right)^2 = y^{5 \cdot 2} = y^{10}$[/tex], we have:
[tex]$$\left(5xy^5\right)^2 = 25x^2y^{10}.$$[/tex]
2. Next, simplify the term [tex]$$\left(y^3\right)^4.$$[/tex]
Raising [tex]$y^3$[/tex] to the power of 4 gives:
[tex]$$\left(y^3\right)^4 = y^{3 \cdot 4} = y^{12}.$$[/tex]
3. Now, multiply the results of the two simplifications:
[tex]$$25x^2y^{10} \cdot y^{12}.$$[/tex]
When multiplying powers of the same base, you add the exponents:
[tex]$$y^{10} \cdot y^{12} = y^{10+12} = y^{22}.$$[/tex]
Thus, the product becomes:
[tex]$$25x^2y^{22}.$$[/tex]
The simplified expression is
[tex]$$\boxed{25x^2y^{22}}.$$[/tex]
