Which expression is equivalent to (16 x Superscript 8 Baseline y Superscript negative 12 Baseline) Superscript one-half?
Negative 4 x Superscript 4 Baseline y Superscript 6
Negative 8 x Superscript 4 Baseline y Superscript 6
StartFraction 4 x Superscript 4 Baseline Over y Superscript 6 EndFraction
StartFraction 8 x Superscript 4 Baseline Over y Superscript 6 Baseline EndFraction
The correct Answer and Explanation is:
To solve the expression ( (16x^8 y^{-12})^{\frac{1}{2}} ), we will apply the rules of exponents. Specifically, we will use the power of a product rule and the power of a quotient rule to simplify.
Step-by-Step Breakdown:
- Start with the given expression:
[
(16x^8 y^{-12})^{\frac{1}{2}}
]
The first step is to apply the exponent ( \frac{1}{2} ) to each term inside the parentheses separately. - Simplify the constant:
[
(16)^{\frac{1}{2}} = \sqrt{16} = 4
]
Since ( 16 ) is a perfect square, taking its square root results in ( 4 ). - Simplify the variable with exponents:
- For ( x^8 ), we apply the power rule:
[
(x^8)^{\frac{1}{2}} = x^{8 \times \frac{1}{2}} = x^4
] - For ( y^{-12} ), we apply the same rule:
[
(y^{-12})^{\frac{1}{2}} = y^{-12 \times \frac{1}{2}} = y^{-6}
]
This gives us ( y^{-6} ), which can also be written as ( \frac{1}{y^6} ) to eliminate the negative exponent.
- Combine all parts:
After simplifying, the expression becomes:
[
4x^4 \cdot y^{-6}
]
This is equivalent to:
[
\frac{4x^4}{y^6}
]
Final Answer:
The expression that is equivalent to ( (16x^8 y^{-12})^{\frac{1}{2}} ) is:
[
\frac{4x^4}{y^6}
]
Thus, the correct answer is:
[
\text{StartFraction 4 x Superscript 4 Baseline Over y Superscript 6 EndFraction}
]
Explanation (300 words):
The problem involves simplifying the expression ( (16x^8 y^{-12})^{\frac{1}{2}} ) using exponent rules. The primary rules at play are the power of a product rule ( (ab)^n = a^n b^n ), which allows us to apply the exponent to each factor inside the parentheses, and the power of a power rule ( (x^m)^n = x^{m \cdot n} ).
First, we simplify the constant ( 16 ) by taking its square root, which gives ( 4 ). Then, we simplify the variables by multiplying their exponents by ( \frac{1}{2} ). For ( x^8 ), multiplying the exponent by ( \frac{1}{2} ) gives ( x^4 ). For ( y^{-12} ), multiplying the exponent by ( \frac{1}{2} ) gives ( y^{-6} ), which simplifies further to ( \frac{1}{y^6} ).
By combining the simplified terms, we obtain ( 4x^4 \cdot \frac{1}{y^6} ), or equivalently ( \frac{4x^4}{y^6} ). This expression matches the third option, making it the correct answer.
Understanding these rules is key to handling algebraic expressions involving exponents, especially when dealing with fractional or negative powers.