To solve this, you need to know three exponent rules:
1) Power of a product
Basically says [tex] (ab)^{2} = a^{2} b^{2} [/tex]. This means a product raised to a power is the same as taking each factor to that power and multiplying them.
For example: [tex] (5a)^{2} = 5^{2} a^{2} [/tex]
2) Product of powers
Basically says [tex] a^{m} a^{n} = a^{m+n} [/tex]. When two expressions with the same base (a) are multiplied, you can add their exponents while keeping the same base.
For example: [tex] a^{2} a^{3} = a^{2+3} = a^{5} [/tex]
3) Power of a power
Basically says [tex](a^{m})^{n} = a^{m \times n} [/tex]. When an exponent is being raised to a exponent, you can multiply the exponents.
For example: [tex](a^{2})^{3} = a^{(2 \times 3)} = a^{6} [/tex]
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Back to your problem:
You are asked to simplify [tex](6x^{2} y)^{2} (y^{2})^{3} [/tex], Tackle it by simplifying both factors and then multiplying them together and simplifying again.
1) First use the power of a product rule to change [tex](6x^{2} y)^{2}[/tex] into [tex]6^{2} (x^{2})^{2} y^{2} [/tex]. Simplify it into [tex]
36 x^{4} y^{2} [/tex] using the power of a power rule.
2) Simplify [tex](y^{2})^{3}[/tex] into [tex] y^{6} [/tex] using the power of a power rule.
3) Multiply the simplified factors from part one and two and simplify using the product of powers rule:
[tex](36 x^{4} y^{2})(y^{6})\\
= (36 x^{4})(y^{2} y^{6})\\
= (36 x^{4})(y^{2+6}) \\
= 36 x^{4}y^{8} [/tex]
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Answer: [tex]36 x^{4}y^{8} [/tex]
