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How to Find the Equivalent Expression for $(4x^{3}y^{-1})^{-3}$
Mathematics
Grade 9 (Junior High School)
Question Content
Which expression is equivalent to $(4x^{3}y^{-1})^{-3}$?
Correct Answer
$\\frac{y^{3}}{64x^{9}}$
Detailed Solution Steps
1
Step 1: Apply the power of a product rule $(ab)^n=a^n b^n$ to expand the expression: $(4)^{-3} \\times (x^{3})^{-3} \\times (y^{-1})^{-3}$
2
Step 2: Apply the power of a power rule $(a^m)^n=a^{m \\times n}$ to each term: $4^{-3} \\times x^{3 \\times (-3)} \\times y^{(-1) \\times (-3)}$
3
Step 3: Calculate each term separately: $4^{-3}=\\frac{1}{4^3}=\\frac{1}{64}$, $x^{-9}=\\frac{1}{x^9}$, $y^{3}=y^3$
4
Step 4: Multiply all the terms together: $\\frac{1}{64} \\times \\frac{1}{x^9} \\times y^3 = \\frac{y^{3}}{64x^{9}}$
Knowledge Points Involved
1
Negative Exponent Rule
The rule states that $a^{-n}=\\frac{1}{a^n}$ where $a \\neq 0$ and $n$ is a positive integer. It converts a term with a negative exponent to its reciprocal with a positive exponent, used to rewrite expressions without negative exponents.
2
Power of a Product Rule
The rule states that $(ab)^n=a^n b^n$ for any non-zero real numbers $a$, $b$ and integer $n$. It allows us to distribute an exponent to each factor in a product when raising the entire product to a power.
3
Power of a Power Rule
The rule states that $(a^m)^n=a^{m \\times n}$ for any non-zero real number $a$ and integers $m$, $n$. It is used when raising a power to another power by multiplying the exponents together.
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