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Simplify the Algebraic Expression $(3y^{4})^{3}+2y^{7}-4y^{12}$
Mathematics
Grade 8 of Junior High School
Question Content
Simplify the algebraic expression: $(3y^{4})^{3}+2y^{7}-4y^{12}$
Correct Answer
$27y^{12}+2y^{7}-4y^{12}$ or $23y^{12}+2y^{7}$
Detailed Solution Steps
1
Step 1: Apply the power of a product rule and power rule for exponents to $(3y^4)^3$. The power of a product rule states $(ab)^n=a^n b^n$, and the power rule states $(a^m)^n=a^{m\\times n}$. So $(3y^4)^3=3^3\\times(y^4)^3=27y^{12}$.
2
Step 2: Substitute the simplified term back into the original expression: $27y^{12}+2y^7-4y^{12}$.
3
Step 3: Combine like terms. The terms $27y^{12}$ and $-4y^{12}$ are like terms, so $27y^{12}-4y^{12}=23y^{12}$. The final simplified expression is $23y^{12}+2y^7$.
Knowledge Points Involved
1
Power of a Product Rule
This rule states that for any real numbers $a$ and $b$, and positive integer $n$, $(ab)^n=a^n b^n$. It is used when raising a product of two or more factors to a power, allowing you to raise each factor to that power individually. For example, $(2x)^3=2^3x^3=8x^3$.
2
Power Rule for Exponents
The rule says that for any real number $a$, and positive integers $m$ and $n$, $(a^m)^n=a^{m\\times n}$. It applies when raising an exponential expression to another power, where you multiply the exponents together. For example, $(x^2)^4=x^{2\\times4}=x^8$.
3
Combining Like Terms
Like terms are terms that have the same variables raised to the same powers. To combine them, you add or subtract their coefficients while keeping the variable part unchanged. This is used to simplify algebraic expressions, such as $5x^2+3x^2=(5+3)x^2=8x^2$.
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